Prophead,
Didi a similar thing myself. No probs flying your other half - the bit of paper the examiner signs at the end of the flight test IS your licence (until the plastic one comes through the post - which takes even lomger than the CAA!).
6 weeks for a visa is certainly do-able, but I would apply now - always leve plenty of time when dealing with US immigration (use search for info on what to expect when down in London town for your interview)
And finally, if you have a connecting internal flight in the US - also leave plenty of time! They don't like foreigners coming in on an 'M' visa to learn to fly! Expect to sit in a room for an hour while they call your flying school (an hour

!) to verify who you are and ensure you miss your connection
Enjoy and good luck
RP