I've been playing around with the idea for a new character, but I'm curious whether I'm going to run into unforeseen problems at later levels. The idea is to take a Variant Human Fighter and pick up the Magic Initiate feat at first level to get access to shillelagh.

I'm going to dump both Str and Dex, and pump up my Wis and make attacks with that. The idea will be to fight with a heavy stick. I'll pick up the Battle Master fighter archetype, which has enough maneuvers that don't require a saving throw to be worthwhile.

The basic background is that the character is a soldier who got nearly killed and is basically physically crippled, but was rescued and trained as a Druid with an Elven teacher. He'll be druid-like flavorwise, but with an emphasis still on melee combat, using his understanding of and bond with nature to pull him through where his body couldn't.

The only things I've found that are going to hold back this character are that he'll be too weak to wear his armor without a speed penalty (so I'll be slow), at least unless I'm able to pick up Mithral armor and that his Battle Master saves depend on his dumped abilities, but there's enough of them that I can just not pick the ones that have a saving throw.

I wasn't planning to multiclass anyway, but I won't be allowed to with these stats even if I wanted to, so that's not an option. This needs to be a pure Fighter.

However, while I'm pretty sure this'll be fine at the first few levels, I'm curious if I'll run into trouble at later levels when playing him. It'd suck to realize that at some level, the idea suddenly doesn't work anymore and I can't advance any further.

Is there any reason why this low-Str, low-Dex Fighter would be impossible to play at a certain level?

You should be able to play this

It is in no way optimal, but it's certainly possible to play this kind of character, and it will provide you with great roleplaying opportunities. A crippled fighter who uses what little Druidic magic he knows to get by where his physical form can no longer help is a great concept.

As you've already determined, having a low strength and a low dex, you are stuck with having terrible AC unless you wear heavy armor, which has a strength requirement unless it's a mithral armor. You will be able to use a shield alongside a club (heavy stick) for a bonus +2 AC, which means you'll be a fairly tanky character if you don't dump-stat your con as well for the theme.

You will also need to use your bonus action every minute to cast Shillelagh or you'll struggle to hit anything, but once you've done that, you're essentially just using a one-handed longsword, which is a perfectly viable build. You'll be weak to dispell magic and anti-magic zones, but in all honesty, if somebody is going to cast that to dispell your cantrip, they've wasted a spellslot.

For your once per day spell from Magic Initiate, you might consider Absorb Elements. Because you'll be dump-statting your dexterity, there's a decent chance you're going to fail dexterity saves as you are not proficient in those, and a well-timed Absorb Elements might well save you 10+ damage from a spell.

For your other cantrip options besides Shillelagh, you might want some way to 'close distance' or attack from a range, for those situations where your movement (especially when you're slowed down by armor) can't get you near enough to the enemy. With a decent wisdom modifier, your Thorn Whip would allow you pull enemies closer reasonably well. Gust might be useful to push enemies over edges from a distance or push unattended weapons around, allowing you to be more useful from a range than your slow speed would normally allow.

So in conclusion, is this going to be the strongest character at your table? Almost certainly not.

Is it going to be unplayable? I don't think so. I think that once you find Mithral Armor or find some other way to get a decent AC, your character will be perfectly viable in anything but a heavily min/maxed environment.

Lots of things going against you...why not Monk?

As a Battlemaster Fighter, you will find lots of hurdles to get past. Its not just having few weapon options, few maneuvers to choose from, or severe lack of armor choices; there is literally nothing that having Wisdom for a Battlemaster will benefit (beyond skills)

However, as a Monk, your AC is increased, as are your Saving Throws. Character features, like with the Way of the Long Death, are greatly improved. The only thing a Monk loses by going this route is some damage on the Unarmed Strikes, which is easily ignored due to your greatly enhanced Stunning Strikes. Imagine having a 15 DC Stunning Strike for 1 ki point? Don't forget that it's simply attached to an attack you've already hit with, and your 5+ ki points refresh on a Short Rest.

So while Fighter has a lot going against it, a reflavored Monk would be an excellent recommended choice. Subclass options that don't utilize much Dexterity include the Ways of the Long Death (as a tank) and the Four Elements (as a controller).

Side Note: Rather than grabbing Magic Initiate at level 1, starting a single level into Druid will unlock the level 1 spell list, save you a feat, or allow you to choose a different race (such as a Wood Elf or Hill Dwarf). Many of the level 1 druid spells are beneficial for a Monk, including things like Jump, Longstrider, Fog Cloud, or Absorb Elements, giving you more versatility in-and-out of combat.

I'm going to frame-challenge a little bit and say:

Who cares if you run into issues at later levels?

You may run into issues later, but that's part of the character. This character may end up with a shorter "life-span" than other characters you've played, but not every character should live forever. Maybe he'll go out sacrificing himself in some way to save his friends. Maybe adventuring will get too hard for him, he'll retire (aka become an NPC) and become a sage that the team goes to for advice.

You have an interesting idea that you want to play, so play it and have fun.

The reduced speed you mention for wearing armor just adds to the flavor. It goes well with the other things the character would be weak at.

Since your concern is about later on, reduced speed is countered by a magic item which increases speed.

Even more important is that you are going with RP over meta-gaming. This is great. As such, I would be talking to the DM about the possibility for ad-hoc benefits.

Maybe you could get the DM to let you choose to do arbitrary but cool and useful things on occasion if you can work them in well and they are awesome; ask if the "rule of cool" can be extended for you.

Maybe you can ask for some customizations to your class. Your concern is about power level later in the game, and you say you cannot multiclass but are being taught by a druid. Why not ask the DM to either relax the rules for you so you can multiclass, or just say "Can I have some low-level druid powers later on just because.

It would be cool, and it would only balance the game, not break balance." If you do this and the DM is hesitant, make sure that you are willing to tone down the power level at any time if DM decides it seems overpowered. And that should be easy to do in-character: DM says "I think we need to re-discuss your guy. He seems to be outperforming others now." You say "OK, he's already got crippling issues, so in-character let's say the powers are causing him too much stress and he has to tone it down if he wants to live." and you scale the abilities back.

Working things out with your group to make the game balanced, fun, and cool is within the rules.