Part of the issue is the likes of Royal Mail are doing their annual £18ph/£32 O/T recruitment drive. Also, ADR licences require periodic renewal, so a C+E ADR driver that hasn't been able to renew their ticket because of TCV isn't going to sit around and wait for the house to be repossessed, ergo a diminishing supply of tanker drivers.
The logistics supplier to Asda and Sainsbugs has gone bust. Probably from paying agencies £100+ an hour for staff. Those staff and the trucks they drive haven't suddenly vanished. If the price of food needs to rise to cover the cost of the supermarkets taking those jobs in house, so be it. (There is a reason why Waitrose runs its own fleet).
The explosion in online retail over the last few years hasn't helped either. When I worked at TNT in Crawley (2004ish), they used to get four standard trailers a night from the hub... Five in the run up to Christmas. Pretty standard at the time.
Currently doing a part time stint at Hermes in Crawley... They're processing 10+ double deck trailers a night. So that's 10 drivers a day, plus another 16 hgv drivers to deliver the contents of those trailers to all the sub depots. If that's typical, then factor that across the courier industry. That's where your driver shortages come from. Not Brexit.
About a third of drivers I currently work with are Eastern European. Most of those that left did so because if TCV not Brexit.
Wages may have risen slightly, but people are still having to work sixty hour weeks to get anything like the reported wages.