Even before Brexit, the UK was suffering from a massive shortage of truckers. Covid and the resulting upset of the transport industry definitely had an impact as well.
However, many UK truck drivers were foreign nationals that all got sent home or didn't believe in their job security after Brexit. Nothing in economics is ever the result of just one thing, but Brexit has made an existing problem much worse.
The difference can be seen all around the UK. Every European country has some kind of shortage in the logistics industry, but only in the UK are the problems bad enough that the army needs to step in.
The biggest, most publicly-visible problem that caused the army to have to step in - the fuel crisis - wasn't really caused by the trucker shortage itself though. It was a media-created phenomenon; breathless front-page headlines about petrol stations running out of fuel caused everyone to go out panic buying, and the infrastructure just doesn't have the capacity to handle everyone filling their tank at once. The actual underlying problems were so tiny that no-one would've noticed them without the media pushing it - a handful of stations out of fuel in the whole country - and as far as I can tell that's what happened in the USA. According to financial publications like Bloomberg and FT they've been having ongoing problems with petrol stations running out of fuel due to a shortage of tanker drivers too, but the mainstream media hasn't covered it so there's been no panic buying, no crisis, people don't even know it's an issue.
What you're saying is not backed up by the numbers. One of the reasons the army stepped in is because our demand exceeded what we could supply with our number of tankers. So we had enough tanker drivers, but not enough tankers. The tankers did not emigrate.
I am Polish. Poland reverted back to COVID-normal a lot earlier, and was later to implement lockdowns, than the UK.
For people on the fence about going back home, lockdowns were a major driver. Additionally, Poland has rejected vaccine mandates, whilst the UK seems on the cusp of implementing them, and requires them for re-entry to the UK.
For people who have experience living under totalitarianism, being forced by the Government to take a medical treatment (often for no purpose since many working-class people already survived COVID) is unacceptable.
The other big factor is that a lot of people who have worked in the UK for the last 5-10 years have now saved up enough money to buy a nice house in Poland. By contrast UK real estate remains continuously unaffordable due to the restrictions on development and continued population expansion from immigration (legal and otherwise).
These trends of de-migration would have played out slowly over the next 5 years but COVID rapidly accelerated them.
However, many UK truck drivers were foreign nationals that all got sent home or didn't believe in their job security after Brexit. Nothing in economics is ever the result of just one thing, but Brexit has made an existing problem much worse.
The difference can be seen all around the UK. Every European country has some kind of shortage in the logistics industry, but only in the UK are the problems bad enough that the army needs to step in.