The pushback lasts 1 minute or until your concentration is broken

The rules

A line of strong wind... blasts from you in a direction you choose for the spell’s duration.

When it says that the wind is blowing for the spell's duration it means that the wind is blowing for 1 minute (or until you stop concentrating on it).

The next line says:

Each creature that starts its turn in the line must succeed on a Strength saving throw or be pushed 15 feet away from you in a direction following the line.

So the line of wind is present for up to one minute, and, as long as that line is present, creatures that start their turn in it must save against its effect or be pushed back. The applies to every turn that a creature starts its turn in it for the record.

I'm afraid your DM is not following the rules as written. I think it is pretty clear that the spell works, and is intended to work, as I have described above.

Common sense

The fact that this is a concentration spell should be a giveaway that some continuing effect should be present. If not, what reason would there be to concentrate on it? What lasting benefit does it provide if not the titular wind?

Does it makes sense to have "strong wind" constantly blowing against people and not have them make saves or have any effect at all? It really doesn't.

Talking to your DM

Whatever you do, you must respect the DM's right and ability to modify the campaign as they see fit. However, you should bring up to them (privately if at all possible) and respectfully ask them to look it over again and see if they might have misread or misunderstood the spell.

If they are still insistent in the previous interpretation, explain to them how badly this nerfs the spell (especially compared to other 2nd level spells). Ask them if they would mind interpreting it your way for a session or two and see if they think it is overpowered (because it is not) or anything wrong becomes apparent to them.

If none of the above works, you may just have to live with it and might consider just not preparing it again if you think it is not worth it. You may also want to ask the DM if you should expect any other changes to rules/spells/abilities to head off any future misunderstandings (thanks @MC).

If there are other serious issues that are seriously affecting your enjoyment of the game you may want to consider starting to look for another group. However, do remember that this is only one spell and probably not worth ruining an otherwise stellar gaming experience for.

Point of clarification: the question as you pose it in your title asks if gust can move "someone" under the effects of levitation, but in the body of your question you are specifically asking about moving yourself. I will focus on the latter.

By Rules as Written? No, absent DM fiat.

The general rule is that spells do only what they say they do, and anything more is up to the DM. The description of gust, which you've quoted, says it can be used to push others, but says nothing about moving the caster.

To put this in perspective, consider that you are effectively looking to use gust to target yourself on the receiving end of the push effect. Technically, targeting yourself might be fair game; the spell says you can target "[o]ne Medium or smaller creature that you choose," and you could perhaps choose yourself. The trouble is, the target is "pushed up to 5 feet away from you," and you cannot be pushed away from yourself. The DM could choose to ignore that bit of illogic, but it would be fiat.

By Rules as Fun? Absolutely yes.

Gust is an underwhelming cantrip. Most of its effects are duplicative of other (better) cantrips -- except for the push effect, which can be achieved just as easily by any character using the rules for shoving a creature. See PHB p. 195. Permitting a PC to cast gust in order to achieve 5 feet of lateral movement while levitating rewards creative play, and is highly unlikely to break the game. (Consider, for example, how difficult it would be to abuse this setup in combat. The PC is expending an entire action for 5 feet of movement.)