If you are planning to move straight from a politics undergraduate degree into sports consulting, you'll need to have something extra in your background to interest consulting firms. Sports consulting is pretty high on most people's wish list, but it's not a large sector, so to stand out from the wannabes you'll need to have done something significant already in sports - Olympic athlete, England cricketer, possibly senior administrator in sporting organisation etc.
Going on to do a Masters might help, but not a lot. There are Sports MScs, and I think Liverpool University's business school does an MBA in Football. A bit of googling should provide you with a list.
I'm inclined to suggest you join a graduate management training programme in a large entertainment/leisure business for a few years (if you can find a world-class employer in this sector), and then try consulting.
As a first course of action, I'd recommend speaking to consultants in sports consultancies to seek their advice on the best way forward.
Here are some firms. The contact names may no longer be current, however....
LEK - Gil Moran. MD Sports Practice
Mott MacDonald - Andy Preece, Director
Neilsen Sports - Glenn Lovett, MD Sponsorship
Simon-Kucher UK - Rosalind Hunter, MD Sports Pricing
The Sports Consultancy - Matthew Wilson, Director