After considering it for a long time, I have finally decided to try DMing.
The campaign I'm planning is entirely homebrew, and, as an aspiring author, I'm going to focus a bit more on lore, plot, and character building than on combat, but there will still be fighting (I've planned at least a few boss battles.)

Three of my friends will be players, but only one has played before. The players opted for a half-elf cleric, a gnome warlock, and a tabaxi bard, and now I'm worried about this party's composition.

Clerics and bards are often healers and usually lack meaningful offensive spells, so I sense they may end up relying on the warlock for offense. Should I be concerned that this party will be unable to defeat foes that a more traditional party could? Should I encourage the bard's player, who upon making her character stated that she'd be okay with playing a different class despite her initial preference, to pick a different class so that the party can engage in combat better?

You don't have a party balance problem unless you create one.

In certain editions of D&D (mostly 4th), there were definitely more stringent requirements for class/party balance. That is not the case in 5th Edition. Any party can be successful, if they play to their strengths. As a DM you always need to pay attention to the types of enemies you throw at the characters.

What's challenging for a melee heavy party may be simple for your caster-heavy party, while the inverse can be equally true. You should also pay some mind to what the players like. Some enjoy mind puzzles, others enjoy intricate tactical combat, some enjoy pure socializing. As long as the DM doesn't put up deliberate roadblocks, forcing the party to us a single solution, your group should be okay.

Character Competencies

You do have some misconceptions about how the bard and cleric can function. They are certainly both capable healers, but they don't have to focus on that. In 5E, bulk healing is often an out-of-combat task: heals-per-round can never keep up with damage-per-round. It can only bring people up who went down and maybe buy a little extra time. Good crowd control is actually more important than good combat healing, and crowd control is something both the bard and cleric can do.

Depending on domain, Clerics can be competent at healing (Life is super good at it), battle (War or Tempest), or even combat spellcasting (Light). All clerics have access to some real gems, like Guiding Bolt (solid damage, and grants advantage to a later attack) and Spirit Guardians (one of the best lower-level area control spells).

Bards can heal, sure, but honestly... there are a lot of classes that can get access to healing. What bards really do is support and control. Don't underestimate Bardic Inspiration; a few points here or there can be clinch. Even low level spells like Dissonant Whispers have good crowd control abilities, causing people to run and provoke attacks of opportunities. Hypnotic Pattern, an excellent lock-down spell, is on the bard list too. Furthermore, any bard can potentially learn any spell in the game - Magic Secrets is no joke, and Lore Bards get it relatively early.

The only one-trick pony in your group is the warlock, and even that's a slight exaggeration. They're certainly less flexible than other spell casters, but warlock invocations can be very powerful. If the player picks the right spells (if it doesn't scale with level, it's probably not a good Warlock pick) they can be a powerhouse.

A Note on Worldbuilding or "No plan survives first contact."

Never forget that it's a gaming group. The DM's responsibility is to facilitate fun for everybody. It sounds like part of your fun includes the world building aspect, and that's great. Just be careful to give the players options; let them explore your world, don't force them down a particular path just because you have a neat bit of lore to share over there. Encourage, but never force.

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

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.

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.