Well, the people I knew at university who went on to become PMs, ministers etc, were all involved in student politics, some of them were party members, and organised the Young WhateverParty at university (and probably at school before that). Student politics was their hobby & their training ground. Proximity to Westminster or wherever, doesn't matter, because it's the networks and training in activism that matters about student politics.
And serious student politicians would attend their local party meetings, as well as the annual party conferences, as well as the annual student union conferences.
I would say that what might make the difference is the university someone attends in terns of elite /ruling class status. Rare that a PM comes from Bolton Poly (although it wouldn't be a bad thing if she did!)
Although, Cat Smith was a student at Lancaster, and now represents that electorate and is, I think, climbing the greasy pole of the Labour Party factions - not as swiftly as Wes Streeting, though they have similar student politics profiles. Difference between Lancaster and Cambridge is my point ... (although Lancaster is a truly excellent university!)