Both of these institutions seem to rank quite high overall, with one or the other ranking higher depending on where you look. Any recommendations when it comes to pursuing a career in Consulting? Are they close enough that I should be focusing on secondary factors in order to decide which one to prioritise? I'd appreciate some guidance :)

Start with the firms you want to work for: which of them has the best network for you? 

Yes. Consulting is such a vast field with probably thousands of firms focusing on various areas / niches. You will have a hard time at the best firms (MBB) if that's your goal and these are the only schools you are looking at. 

Pick a handful of realistic firms that you would like to work for and then check LinkedIn: are there grads from either of these schools working at those firms? How many? 

Would say Warwick. They should have more alumni working in those fields.