After signing the contract, my start date was delayed by a month to 1st February but I still haven't received a contract with the new start date. I mentioned this when I had lunch recently with my future boss. What should I do?
Given that February 1st is next week, they probably won't send it to you, they will just have you sign it the minute you show up.
More likely it will not be a brand new contract (that all parties would need to read and make sure no side slipped something in besides the start date), it will just be a single page with the official letterhead as an addendum to the contract saying something like "the start date was moved to February 1st" and you both sign it.
If they do not, because they don't deem it neccessary, just tell them you need that because you received unemployment benefits for January and you don't want to get in trouble with any German buerocracy over getting benefits and being employed, even if only on paper and for no money. Germans know that we don't want that. We don't want trouble over paperwork. They will give you that sheet to sign, probably two of them, so both you and the company have their own copy with two legit signatures.
If this job is remote, just call the HR and tell them. Paperwork is their job, they will understand.