Is the construction time specified in the prefabricated house contract binding?

In the end, it doesn’t really matter. It becomes easier with a specific contractual penalty.

Otherwise, § 286 of the Building Code applies (no legal advice!). The company falls into default and becomes liable for damages. Of course, everything must be reported correctly by you then.
Then, for example, they have to pay the following things (I copied this from the internet because I find the list useful):

[*]The rental costs for the previous apartment incurred during the delay period,
[*]any additional travel costs arising because the move could not yet take place,
[*]any hotel accommodation costs or interim storage costs for the furniture, if the buyer or builder has terminated or sold their previous apartment in reliance on the agreed completion date and must vacate the previous apartment by the promised completion date.
[*]Financing costs, such as commitment interest, because the last payment installments could not yet be drawn from the lender due to the lack of completion. In addition, interest on the already drawn loan capital must also be considered as damage for the duration of the delay. Basically, the repayment phase of an annuity loan only starts with the full disbursement of the loan amount, so the repayment period usually extends by the duration of the delayed completion of the construction project.
[*]Attorney fees can also be reimbursable if the buyer or builder hires a lawyer after the developer or general contractor falls into default.
[*]Finally, the buyer can also claim compensation for loss of use of the apartment not made available on time. However, it should be noted that this compensation for loss of use must naturally be offset against any concretely claimed rental damage for the continued rental of the previous apartment. Thus, no cumulative damage claims should be made in this regard either.

Whether it might then be possible to offset this with the payments still to be made or the final payment can better be assessed by a lawyer.

Thanks anyway for a rough direction.
It is definitely good to know that there are fallback regulations somewhere if no contractual penalties are defined. Now I just have to find the general terms and conditions again and sift through them, but at least I do not assume they say "If the provider does not fulfill their contractual obligations, you’re out of luck. Too bad, so sad."

It will also be interesting to see who we can talk to specifically about the whole thing. We are not intent on a legal dispute at all costs. We can also accept that the construction is delayed. We just don’t want to be liable for costs caused by a delay that we are not responsible for and that should not exist according to the contract.

Otherwise, § 286 of the Building Code applies (no legal advice!). The company falls into default and is liable for damages. Of course, everything must also be correctly reported by you in that case.

Since no start date is fixed, no default can be asserted for a long time. Planning can take years with the general contractor's good will ;) The statement "The delay is due to a high volume of contracts, according to the consultant" might possibly get you out of the contract but will not get you the house any faster.

Since no start is fixed, no delay can be claimed yet.

The start is the problem. Approved building application. Until then - especially if the construction company also prepares the plans and the application - it can of course take time. But the construction company is not allowed to simply botch that either.

and I don't really assume that the BT is speculating on falling prices...

Since no start is fixed, no delay can be claimed yet. Planning can take years with the good will ;) of the general contractor. The statement "The delay is due to a high volume of contracts according to the consultant" might get you out of the contract, but it won’t get you the house any faster.

The start is fixed as the approval of the building application. The first installment is also due at this starting point. If the company postpones this point, it is unpleasant, but since we only sign the loan at this time (as actually intended by the builder), the interest on the provided loan is not a problem as long as the provider adheres to the contractually agreed timeframes from that point on.

Then everything is fine. Hopefully the GU sees it that way too.