My ISP just went bust - what now?

Factual Questions
1

Or at least it looks that way; I can’t make any contact with the servers for squareinternet.co.uk and they are not answering the telephone.

Does this complicate the matter of transferring my domain names to a new ISP? what fees will I be expected to pay?

2

I don’t think it complicates this particular issue. Normally, you inform your new ISP, and they get in contact with your old ISP to arrange for the transfer.

In fact, there’s no transfer. What happens is just a record in the DNS database gets updated.