BackPacker, I'm not sure your solution works. Reason being, that
AFAIK the CAA (UK) respond to the query of another CAA (the FAA in this case) directly and will not give you the required docs. Otherwise this whole procedure would be a farce. I got caught up in this situation last year, albeit in Africa. Sorted it out in the end, but the CAA quote 10 working days for dealing with a request. They will, however,
expedite if their reply is required for training!.
There may be two solutions here:
1) Speak with CAA and FAA and see how the verification process can be expedited, then get a piggyback FAA PPL
2) (SoCal or someone else correct me if this is wrong) Do all your training and the written test in Europe, then only do the checkride in the US. Contrary to JARland, under FAA rules YOU are PIC during the checkride, so technically this should not be considered 'training'. Again, don't take my word for this, but check.
I know one FAA CFII on the continent, you could get your sign-offs from him. PM me if details needed.
HTH