Ohhhh such an exciting trip to plan, Jacinda! I'd love to travel Europe too, one day. When I was in UK for 10 years, I was just a poor student with budgets, so I couldn't really travel.
In London/UK, I loved to stay in Travelodge, as the room is spacious, clean and affordable.
I am not sure if they still exist though, but it's worth checking.