I've never been through the process of obtaining a piggyback PPL myself, but what I've seen is that on the US side, things are almost instantaneously. It's the UK CAA which allows itself 14 days or so to issue the verification of authenticity to the FAA.
All in all, worst case should be around three weeks, I guess.
Furthermore, for a piggyback FAA PPL which you're going to use for hour building or holiday, you don't need TSA clearance and/or an M-1 visa. If you want to obtain a standalone FAA PPL you definitely do need TSA clearance, and most likely need an M-1 visa.