While other universities don't have the same tradition of formals, there will be occasions to dress up whatever university you go to! Lancaster, like the unis you mention, is also a collegiate uni. Unlike oxbridge and Durham however, Lancaster is a relatively new university (built in the 60s) and our colleges are all integrated into the main uni campus. So while we don't have banquet halls and robes, we do have college traditions and events throughout the year. Each college host a Winter Ball and an end of the year 'Extrav'. Winter Balls are usually held off campus in a fancy hotel with a formal dress code and sit down meal while Extravs are more like a party with each college choosing a theme with live entertainment, food, drinks etc to go along with it. Lots of societies at Lancaster will also host formals/ balls alongside college ones. These are always a great excuse to dress up, have a nice meal and spend the night with good friends. Some more academic based societies will also hold formal networking events where they invite external guests.
While I have only experienced student life at Lancaster, I have heard from friends at other unis that this sort of society event is common across all unis. Where there are students, there will be excuses to dress up! If you're looking around unis to apply to, it's always work popping into their SU (or looking on their website) to see what societies and social activities they have to offer. University is at its core about education and accademics, but it is also really crucial that you feel welcome and enjoy your time there.
I hope this has helped to answer your question, I'm sure other current students will come on and share their experiences Yasmin (Lancaster University student ambassador)