Easy assumption that. My observation following my recent brush with the NHS is more nuanced. I didnt make a list, but in the ICU, (higher level of training than wards), the nurses were probably 50/50 Brit/EU. One was ex military, who got bored so upskilled, many of the EU ones moved to the UK with partners, many who were in more senior positions in the NHS, some had studied medicine in the UK, and stayed.
As our treatment got less serious or intensive, the balance between UK and non UK staff was marked. More like 80/20 in favour of Brits.
And as it got beyond medical to diet and physio, we had one non Brit staff. (Kiwi, married a Brit).
Now it was one experience, over nearly 2 months though, in one hospital in a very cosmopolitan city with world class universities and hospitals. So it may have been a fluke. Dunno.
It was interesting watching the melting pot of humanity all doing the same thing; caring for fellow human beings.