A friend called AMHA last week to ask about her paperwork which was sent in mid-December (a number of foal registrations, some transfers and a couple permanents. She was told they were still working on paperwork they had received in December, but her paperwork was complete and had been mailed the end of January. She received it this past Tuesday. So, sounds like it's about 6 weeks at the moment.
I've recently had AMHR paperwork coming back about 3 weeks after mailing--that's pretty fast for paperwork being mailed from/to Canada--I've had it take 4-5 weeks, and once 6 weeks when I sent something in at what was probably a very busy time of year--Nationals time. so, it varies.