The BASA is there, ratified, and the politicians have had their say (which is why it took so long). It's now up to the FAA and EASA to agree an Annex on licensing.
... with "input" from vested interests on both sides
As already mentioned, the FAA has little to lose by supporting this. The USA already has a very good system, and it allows anybody to fly an N-reg plane, worldwide, on any foreign license/rating, so long as it was issued by the owner of the airspace (FAR 61.3). It also allows all foreign training, in any certified aircraft, with any correctly rated instructor, towards any US license or a rating, right up to the ATP (with some trivial exceptions). This is an extremely generous policy which has made it much easier for non-US pilots to get US papers because their European training has been fully utilised.
But Europe has a lot to lose. The pilot training and component / airframe / modification certifications systems hang largely on protectionist and job-creating practices. The FTOs are scared of hordes of highly cost-sensitive ATP cadets clearing off to do FAA CPL/IRs in the USA. So it is at this end one can expect big trouble.
I could be wrong, but only if the EU gets a big enough prize elsewhere. But I can't see any prize to be had in these areas.
If we do get a bilateral agreement then, as I've said before, it won't be anything like a straight paper swap. It may be stuff like acceptance of US training logbook entries...