The DM will need to adjudicate
The rules do not cover this specific situation, so the DM will need to adjudicate.
The true polymorphed elemental probably doesn't stick around.
In game terms it is reasonable that this won't work. While "true polymorph" changes a the target's "game statistics", it does not change its fundamental essence. It is still itself, deep down. Probably. After all, game statistics are an abstract representation, and do not account for everything. Things such as the soul, family, current location, origin, are not accounted for in game statistics.
Transforming it into an object instead of a creature doesn't change things.
However, there's enough ambiguity in the spell for interpretation and argument, so the DM will need to adjudicate.
Is a true polymorphed elemental friendly?
The conjure elemental spell says:
The elemental is friendly to you and your companions for the duration.
The DM will need to determine if the elemental is friendly beyond an hour.
It's reasonable that even if true polymorph keeps the elemental beyond an hour, it's no longer friendly, since the conjure elemental spell ended. However, there's no reason to believe it becomes hostile, either.
Contrast with planar binding:
If the creature was summoned or created by another spell, that spell's duration is extended to match the duration of this spell.
Planar binding explicitly extends the duration of conjure elemental. True polymorph shouldn't.
A friendly, permanent elemental definitely shouldn't work
This is game-breaking if not checked. At least, it's game-breaking if the elemental is friendly. If the elemental is indifferent, then it's not particularly game-breaking, at least in this instance.
There are many places in the rules where, with the right interpretation, the game is broken, and this is one of them.
The player is probably hoping to find leverage to gain a permanently loyal minion; that's okay, some players and tables like trying to exploit the rules.
If allowed once, the caster gets a permanent minion, and the player gains an extra action or attack. If allowed multiple times, the caster gains many minions, as many as time allows, and the game is broken.
The rules-as-written interpretation
RAW, it's ambiguous. The wording of conjure elemental supports that the elemental is neither friendly nor hostile if the spell runs out and the elemental is still around.
The effects of true polymorph on the elemental are ambiguous at best.
You can find endless places in the rules where the rules don't account for every possible situation. That's actually in the rules, in the introduction to the DMG:
The rules don’t account for every possible situation that might arise during a typical D&D session.
This is one of those situations.
So, the actual RAW answer, is, the rules don't say, it's up to the DM.
Running the game
It's the DM's job to shut down or limit game-breaking actions. The DM is in charge, not the rules. The DM needs to keep the game playable.
A rules lawyer may argue with you. If you enjoy it, great, argue away. If not, game balance and playability are a perfectly valid reason to disallow something.
You do not need a rationale for this. You can just disallow it.
The character can try it
You are also free to rely on "How to play":
- The DM describes the environment.
- The players describe what they want to do.
- The DM narrates the results of the adventurers’ actions.
You can certainly tell the player to try it in-game, and then narrate that it fails. You can have this conversation out-of-session.
Ruling or table rule
It is perfectly reasonable to simply disallow this as a table rule.
It may help if you explain to the player that this can't work, because if it works for the player, it works for everyone else, and the world is full of permanent elementals.
Allowing it once
There may be circumstances where you may with to allow it once. If you want, you can rationalize this in game. You can allow the character to discover knowledge that suggests it can happen under a certain set of unique conditions, for instance.
You can also simply tell the player that it only works once, because that's how magic works. The wish spell does not give a rationale for the lifetime limit, it just imposes it. Similarly, you can allow the player to have this one time only.
My experience with summoned elementals
You don't need this trick to break the game with summoned elementals (or other creatures). If the DM allows planar binding to work on elementals then a character can easily gain a small number of summoned elementals bound for a few days. That's more than enough for the character to have many minions attacking per turn. We experienced this in a game I was in. The consensus amongst the players was, it wasn't fun. We agreed to limit such conjured/summoned minions in number and frequency. It worked much better after that.
If you are the player
You might want to work it out with your DM ahead of time, so that you don't use up session time examining the subject in detail. Furthermore, most DMs appreciate not being surprised at the table with these sorts of shenanigans.