You might find it easier to go through the process in a more flexible EASA country (eg Poland) and then do a "change of state of licence issue" to the UK to get yourself a UK-issued licence (if that is what you are after).
I have looked at that route as an option to migrate from an FAA to EASA and it all looks quite feasible and less expensive than doing it in the UK. I couldn't comment on how that would work with a jet type though. I guess what you could do is do the conversion on a twin piston type and then EASA will recognise an ICAO type rating if you have >500hrs on type?
I'm not an expert on this I'm afraid, I can only offer the route I have looked at for myself.