Other have chimed in with esports specifics.
There are international students decide they want to try for a US school and get on the horse at a very late stage. It can be done, but its not easy (you will be cutting some corners). Some schools do have rolling admissions past May 1st…usually its schools with low yield and spots to fill in their incoming class.
The crucial step in all this is the visa application; I would work backward from that time frame. From my understanding UK as most European cities is pretty quick for student visa applications, but I would still give myself at least a six week window to complete that process. To start that process you need to be admitted. Also its important to have your paperwork ready to go, particularly on the financials (scholarship plus personal funds for all four years). That doesn’t mean you’re going to tap into those funds as an obligation but you need to show proof that you can cover the cost of the education. We have kept excess funds in our accounts in order knowing that we will have to prove this in the next coming months.
As some have pointed out you may not be able to visit all the schools (or even any of the schools…not uncommon for internationals). Youtube tours and videos are your friend…it wont be close to the same thing but much better than going in blind. If you visit, I’d give priority to the most likeliest place. If you’re visiting more than one place 1-2 nights should be plenty of time. Also, Id highly suggest talking to current students, hopefully internationals so they will give you a fair assessment situation.