A Cleric can easily stand in for a front-liner ("tank"), especially if they choose a Domain that grants Heavy Armor Proficiency. The Cleric also has several combat spell options, that expand even further if you have the Xanathar's book (My Life Cleric is a ranged menace with Toll the Dead and Guiding Bolt). The Warlock has blasting and a lot of arcane tricks handled. The Bard is a supporter and back-up healer, just in case. While their exact choices on subclasses and such will matter, this is nearly as perfect as a three-member party can get.

Specifically re: changing the Bard's class - the Bard is the party's only specialist right now. Neither the Cleric nor the Warlock is exactly drowning in skill and tool and proficiencies. Unless the Cleric and Warlock can handle all the lock-picking, sneaking, etc. with their magic, the Bard is probably rather important in this party. I strongly recommend simply suggesting to that player that they choose their skills wisely to cover their allies' weaknesses.

It depends on the subclasses

In my experience at least half the party should be frontliners.

In other words, if there are no tanks dedicated to keeping enemies away from the ranged attackers, everyone will be in melee.

This would especially hurt Warlocks specializing in Eldritch Blast¹.

So the main question is not race and class, but the ability and willingness of two players to stay in melee.
All of these classes have melee oriented subclasses (Cleric Domain, Bard College, Warlock Patrons). If they pick the melee options, the party will be fine, without extra effort from your side.

Of course you can keep even 3 Lore Bards alive, but it requires special encounter planning.

Optimal subclass distribution

It is hard for Bards to achieve a decent AC. Valor Bards do get shields, but they have problems casting in them, unless they take Warcaster. And it only kicks in from level 3.

Hexblades are very viable in the front line, but they can be quite complicated for a beginner.

Clerics in Heavy armor do just fine in the front line.

I would go Nature Cleric, Valor Bard², non-Blade Warlock.

1) Constant disadvantage for your main combat activity is bad for your performance

2) It might look I contradict myself here, but other Bards do even worse in combat. Unless they take 2 levels of Warlock, which I do not recommend for beginners

Adapt your game sessions a bit to the players, since it's a small group

My question is, however, would this make a good party?

TL;DR: yes, Cleric, Bard, Warlock is an OK set up

Three points that I recommend you reconsider before you start.

1. Author and plot




2. Player class change before play starts




3. Perceptions on bards and clerics

But first: Party size

The game's basic design is a four character party, though I've seen "three to five" as a nominal basic party size tossed around. I've played with three quite a bit, and with five. The difference during an encounter is noticeable. If your players are not optimization driven, the action economy can get a three player party into trouble quickly, particularly when the dice swing against them. If you are heavily into exploration and social encounters, it won't matter as much and each player gets more spotlight.

Author with a plot - proceed with caution

If your players don't mind a bit of railroad riding to keep the game moving, that's fine, but be aware that it is the players' decisions and choices, and those consequences, that drive their fun. Even if you have certain nodes and BBEG's to overcome (the opposing forces in the game world) trying to get the players to fit into your plot can create problems.

Class choice is up to the player - let that be their challenge

I've spoken to my Tabaxi Bard, and she's open to changing her class, as she doesn't really know whether or not she likes D&D yet, but I also don't want to make decisions for her.

New players are confronted with the tyranny of choice. They are the kid in a candy story with only enough money for one candy bar. The use of NPCs, or hired guards/specialists, is one way that you as the DM can offer the players an option of getting the help they need from the game world to overcome various challenges.

• 
  

  This approach is modular: as each challenge or situation arises, they
  may need to seek different kinds of assistance or specialists. It
  can also open up a lot of fun role play opportunities when the PCs
  negotiate for services and access specialists, or as they negotiate
  for/appeal for assistance from various NPCs in the game world. The
  latter process can help them feel that your game world has more depth
  as they form relationships with NPCs by doing that.

  
  
  

  For new players, leave class change options open: if after a few sessions they find that they are not happy with their character class let them swap to a different one. In official play, they allow this in Adventurer's League up to fifth level.

  
  

Keep that option open in case you need it, particularly for new players.

• 
  Experience. Even experienced players may want to change PC class. I am in a tier 3 group (all PCs are currently between level 12-15). Two of the players (since I joined the group well over a year ago) have let go of a 12th, and a 13th, level character and begun a different character at 11. They join the group during
  the adventure. One of them dropped a wizard to roll up a ranger. It's going fine.

"Bards and Clerics are more like healers" assumption

Not so fast. My Tempest domain cleric in our first campaign was a front liner. So too was my original Life domain cleric (who died) - RIP Korvin Starmast. Bards have a wide variety of ways to contribute to a party. Drop a few of your assumptions and Let The Players Play. Let them find out how to get the best synergy out of their character skills and features within their little team. That's part of the fun.

The Lore Bard can start with some key skills that are usually used by Rogues; (thieves tools, stealth, persuasion, deception, acrobatics). In a small party, having a bard "skill monkey" can be very useful. Song of Rest and Bardic Inspiration are also great party assets.

Long Term considerations

bards are often healers and usually lack meaningful offensive spells

Not in my experience. Our first 5e group had a College of Lore bard and a Tempest domain cleric. We never felt at a loss for offensive magic.

Lore Bards get significant benefits from magical secrets; they can hand pick any spell from any list to fill in offensive, control, or utility. Clerics get spirit guardians at level 5, already have guiding bolt and Inflict wounds at level 1, and get things like earthquake at level 15. (I've seen that spell wreck a frost giant stronghold.) A tempest cleric gets call lightning at level 5. If you have access to Xanathar's Guide to Everything, there are some more choices that you may want to add to their spell lists. If not, the PHB spells are still good enough.

Your whole group is on a journey of discovery; your players are the stars, not your plot.

Mechanical point: use short rests!

Pace the game so that the PCs can take advantage of the short rest recharge of various class features, to include the use of Hit Dice for healing.

See if you can find a fourth player

That may help round out your group in a different way.

Party composition does not matter as long as the GM takes their limitations into account.

Would a Stone Golem (CR 10) make a good fight for a party of 4 level 1 characters? NO! They would all die, BADLY. How about for a party of 4 level 20 characters? That Golem is going to be turned into sand.

What if you only have a single Level 20 fighter? The fighter will probably still win.. but its a much closer fight. All challenges are relative, whats hard for one group will be easy for another. Level is one broad marker, but composition is another. If a party has no way to find traps... traps are always going to be a problem. As a GM you wouldn't send a CR10 monster after a group of Level 1s. If they have no trap finders, don't put killer traps in their path.

Just recently I was in a part of 2, a rogue and illusionist. We had no ranged DPS, we had no front line tanks, we had no healing. Combat was very very scary to us and we ran away from 90% of the fights that presented themselves. The other 10% of the fights were us ambushing the captain of the guard, in the Privy, weaponless, and unable to call for help.

Our GM got slightly frustrated from time to time because he had several grand quests out in the wilderness, cleaning out monsters from abandoned ruins. To which our party of 2 took one look at and said "NOPE!! We like our warm comfy beds at the inn Thank-you-very-much!"

From then on, we became respected business owners by day, and ruthless vigilantes eliminating the corrupt elements of the city by night.

This worked because we found quests/stories that matched our limited proficiency set. If we had been, say, a fighter and a cleric, we would have found vastly different stories and adventures because our skill set would have been totally different.

Run session 1 and 2 as exploratory sessions.

Something I always do as a GM for session 1, and sometimes 2 and 3 is I deliberately put my players/characters in as many different scenarios as possible. I give them a good vs evil fight, I give them to option to get into a fight for money, I give them a social encounter. Once they have had a chance to see what they can do, I start offering them options. Resolve this issue using social tricks, OR resolve it with combat.

Track what the players were able to do Mechanically well (and poorly), and what they Enjoyed (and disliked), and fine tune your Stories/Quests based on that.

If they liked combat, but were fairing poorly, use lower CR monsters or give them more chances to take short and long rests. If they were enjoying the political, add more social possibilities.

If the party is having problems in some combat encounters the DM could introduce an NPC to cover their weak areas.

In one game I was part of the group had a dog. This wasn't a ranger or mage familiar, just a trained dog. With levels in fighter. And an INT score higher than some Barbarians. :-)

That dog was a melee star. Not only because he could grapple and trip but because for whatever reason his dice rolls were amazing. Whoever rolled for the dog seemed to get critical hits far more often than should have happened.

Everyone had a soft spot for the dog. He got rescued first every time if he needed healing and everyone saved up buff spells for him.

A less complicated solution

Why don't you allow the party to hire an NPC hireling (like that aged grizzled warrior they meet at the pub)?

The NPC can be a great font of quest wisdom when you need to get things moving, and you can fudge rolls for it when the party needs some help (the warrior crits on the enemy that has the warlock and cleric held releasing them from his evil spell). If anyone dies they can elect to take over the NPC if they want.

Just remember though it is their story - not the hireling/yours - so always let them be the heroes.

Enjoy the roller coaster ride that is DMing

Test the waters

There are too many variables to count to see if any given party can accomplish any given goal, however D&D isn't a game where there can be 'failure' as such, so you can easily test the waters.

I personally in your situation would voice my concerns to the players, framed in a way that as a new DM you don't know if your campaign suits the players, not the other way around. Then just see what happens.

Someone dies? Whole party dies? Roll up some new characters and keep going. Honestly that would be a very good experience for all of you because eventually it happens to most people, and getting it out of the way early on is a good learning experience.