Finding a part-time job for a university student in the UK is not difficult. I bet that wherever you live, it is much harder to find a job as a university student.
The competition is indeed high, but I am sure you will be able to find a job if you apply for plenty of jobs.
Also, there are other jobs university students can do, not just working for fast food restaurants, e.g. waiter/waitress, receptionist, call centre worker, etc. Internships are good too.
You might want to get a job or volunteer where you live (if you do not have one or if you do not do that already) so you have some work experience, which would make you a better candidate than those fresh out of college/sixth form 18-year-olds who have no work experience.
As for switching from part time to full time during the summer, I doubt that would be a problem. You will obviously have to let your workplace know in advance, but I do not think it will be too troublesome.