For CS, Manchester’s one of the best in the UK, maybe even in the world. Southampton and Newcastle also has a great CS department, and I’m sure Nottingham’s good enough too. Nottingham’s great in engineering and pharmacy, but not sure about the CS department as I don’t know anyone doing CS there and never really heard anything about them. But it’s in the Russell Group, and most universities in the RG have similar courses so I wouldn’t worry about the teaching quality.
Especially since you’re an international student, I recommend Manchester. Manchester has the best international reputation, and it’s teaching / research quality is extraordinary. I’d say Manchester > Southampton > Newcastle >= Nottingham, especially since you’re an international student.
However, if you don’t care about the international reputation and are planning on staying in the UK, I’d say go for the city you’d like to stay in. Manchester’s a very busy and urban city, so there will be whatever you want to do there. Southampton’s way more relaxing than Manchester since it’s near the shores of South England, but it’s one of the biggest cities in South England so you’d find most of what you want or need there. Nottingham’s a student city, so there are plenty of students to hang out with and you’d find everything you need as a student including night life, study centres etc. Not sure about Newcastle but judging from what my friend says it’s a student-friendly city.
If you can, visit all of those cities and feel the vibe of them and decide which of them you want to go. Good luck!