There is one part of the advice on the CAA website that is wrong:
Please note that German pilots’ licences are fully valid in all EASA member states.
A German licence is not valid for flying an N reg aircraft anywhere other than Germany.
So, you cannot use a German issued EASA licence to fly an N reg in the UK.
You cannot convert a German EASA licence to a UK one.
The only way is to get a US one issued either standalone or on the basis of the German EASA one.
This just happened to an Italian friend and it took weeks and quite a few £100s to resolve.