The ONS have released their quarterly migration numbers
Net migration -49k to 273k to year ending Sep 2016
Immigration -23k to 596k
Migration +26k to 323k
EU to UK migration -6k to 165k
Non-EU to UK -28k to 164k
Brits emigrating 128k unch (50% work related)
EU citizens leaving UK 103k vs 85k prior (43% work related)
Non-EU leaving UK 96k vs 84k prior (69% work related)
The ONS note that most of the change in numbers here are not statistically significant. For example, on net migration there's a confidence level of 95% & +/-41k.
The long and short of it is that European citizens aren't fleeing in droves and UK folk abroad aren't packing up their wagons and streaming home. That all might change of course, depending on the outcome of Brexit but even then, I still don't see any mass migration moves. It's likely people will have to just fill our some type of backdated immigration forms or register.