Good news is that neither domain is going to expire for a few months at least so you’ve got time.
So you transferred them to Fasthosts with the developer but did he have to pay for any renewals and do you owe him any money? TBH, if you owe him, it’s not unreasonable he wants his money before giving you the domains. Anyway…
The easiest thing is to contact Fasthosts and prove (with a copy of your passport etc) that you’re the owner and assuming there are no outstanding invoices should let you transfer out. If not you need to speak to Nominet and ask them how to proceed. This sort of thing is VERY common so they have processes and in theory, it’s up to the developer to sue you for any unpaid invoices, not for Fasthosts to deny the owner their domains. I work for a registrar and we’ve done this a lot over the years.
Finally, try logging into this page with your email address because if he didn’t change the address on the registrant details, you can log into the site and do the tag change from there.