Behavior/Consequences towards Construction Company

In writing, we have the construction contract with a latest construction end date of 12/2022,

Then the GU still has almost half a year time. Everything is fine and on schedule. That doesn't help you right now, but you can only do one thing for now. Talk to the GU. Everything else makes no sense, because a) the GU is still fully on schedule and b) you therefore have no claims until the end of the year. If you move out of your apartment earlier, it is - I hardly dare to say it - simply your personal and self-made problem.

The agreed and signed construction schedule is from May this year, so that counts for something, right?! Especially since there are no worst-case scenarios (like lockdowns), the completion according to the contract would have been no later than the end of 2022.

We signed under the condition that it must be completed this summer (there is also an email about this, which once again went unanswered). We’re not moving out for fun – but because the contract ends.

It’s almost misleading.

However, the contract is binding and not the construction schedule, unless the contract explicitly states that the end of 2022 only applies under condition XY and otherwise May 2022.
You will not be spared having to see a specialist lawyer. Only he can assess your contract and your options.

Presumably, the contract is structured so that 12/2022 is binding, especially since there are delivery difficulties and therefore deviations from the construction schedule are justifiable. In this respect, having a lawyer involved from the start would have been more sensible.

Originally scheduled for June 2021 it said May 2022

In writing, we have the construction contract with a construction period ending at the latest 12/2022

What do you have written in the contract regarding the handover date?

unless the contract explicitly states that the end of 2022 only applies under condition XY and otherwise May 2022.

Presumably the contract is written so that 12/2022 is binding