How do you decide between two PhD offers?
I am answering this on the basis that I was making this same decision myself a year ago. 

Firstly, the LRI will be joining the Francis Crick Institute next year as it opens, which may end up being esteemed in it's own right if all goes according to their plans. It depends what you want to do after the PhD, if you want to stay in academia and cancer research, then it is the supervisor's esteem that is more likely to help you in the long run, as well as the impact of your papers. The label on the institute makes less difference as you go on (within reason of course).

Have a look at the recent publications of the group, maybe see how many have PhD students on good author positions in good journals; that's what will get you a post-doc ultimately.

The two being 4 year funded, the London one may be better financially overall. Integrated PhD programs (where you have 3 small projects in the first year, getting an MRes, then pick the final PhD project for 3 years, I presume this is the case for Oxford) in my experience tend to be more trouble than they're wort, unless you're not sure on picking a definite research project yet or don't have a Masters. 

The issues can come from firstly, not getting the project you were hoping for after the first year, after the MRes, but now you're tied down to Oxford (may not happen, but does all too ofter) and also you now have 3 years to do your PhD, not 4. Why waste time on multiple smaller sub-projects at the start if you have a clear idea on what you want to do for your final project? Of course, the corollary here, if you don't have a clear idea exactly what you want your PhD to be on, then integrated programs can be a godsend.

My best advice on the rational decision front is don't. Relationship with group and particularly the supervisor is the main factor in a PhD, with a bad supervisor able to kill your career dead in the water, intentionally or not. 

If you have a passion for the research, and get on well with the group and supervisor then I would definitely go for that PhD. Lack of these both for 4 years will likely mean you either drop out, or have a horrible 4 years and then hate your research. 

I went for the lower paid, lower ranked Uni offer, and have no regrets. In fact, the higher paid one was industry funded, and they've since gone under, so I dodged a bullet there.