My perspective is that EU students are not hugely significant income stream for most British universities, particularly at undergraduate level. In fact, they are perceived as a bit of a drain because they pay home fees but tend to require more support due to second language issues. Where universities appreciate EU students tends to be at postgraduate level, particularly taught programmes, where said programmes can be designed specifically for a certain EU market. International undergrads, however, are worth their weight in gold. The uni I work for would far rather recruit twenty internationals than forty EU students.
In fact, if we left the EU and EU students had to pay international fees, then I think we'd probably be a lot more active in trying to recruit them. Again, studying in Britain for an EU student is already significantly more expensive than studying in, say, Germany.
I don't know about other universities, but EU funding is not that significant where I work -- most of our funding comes from private enterprise, state research councils, the UK government or other national governments directly, or charitable trusts (Nuffield, Rowntree etc). I can't actually think of a research project that was funded by the EU in my faculty, and we tend to have a lot of successful proposals every year, and we are awarded multi-million pound grants. When I do see EU research affiliation, it is tends to be through a European wide research network and I am always a little bit dubious about the actual worth of some of these projects and who is actually paying the bills.
The whole EU exchange Erasmus thing? In my experience, it doesn't really work for our undergrads. Many of the European exchange universities we have arrangements with, with the exception of some universities in the Netherlands, require the host language to fluency and our home students just don't speak a European language to that level unless they are bilingual or language students -- and their year out tends to already be built into their 4 year course.
So most of our non-language home undergrads want to go to Canada, the States, New Zealand or Australia, and these exchanges are massively oversubscribed, require an overall first in your first year, work out to be shockingly expensive, and places are allocated by lottery. They are also a hellova lot more prestigious and give the wow factor to a CV.
No doubt other academics will feel differently though, but then one of my primary concerns is that I don't think we are recruiting enough home students to postgraduate STEM programmes and this will cause a massive economic problem for Britain in the future. The students in our postgrad engineering department are almost entirely Chinese, and they go back to China once they have finished their studies. We are pretty much educating another country's youth and ensuring their country's future economic success while ignoring our own needs. It's very blinkered.