Same works the other way. English is a de facto lingua franca. Migrating between different countries within Europe comes with challenges because many countries still stubbornly implicitly require immigrants to know the native language. With the UK in the EU it was the path of least resistance when migration was considered temporary.

I think this is more psychological than factual.

"The number of UK students moving abroad for the whole of a degree did not change in the same way – remaining relatively stable over the past decade. This is perhaps unsurprising given that very few UK students study abroad for the entirety of a degree anyway (particularly in Europe), and those that do so are unlikely to have been severely disadvantaged by the financial changes wrought by Brexit: although they now have to pay ‘international’ fees in Europe, these are typically less than the ‘home’ fees they would have paid in the UK." [1]

There is also the issue of speaking a foreign language well enough to follow an university course as they are usually taught in the local language. Same goes for work.

[1] https://ukandeu.ac.uk/student-mobility-post-brexit/