Honestly the university you attended to do your undergrad does not really have that much of an impact on Oxbridge admissions for postgrad (unless you went to Oxbridge to do said undergrad, in which case you may receive a modicum of preferential treatment). If you are planning on coming to the UK to do an undergrad in the hopes of it being beneficial to your Oxbridge application, I think you may find it to be a bit of a pointless exercise.
If Bristol and York are in your sights, I would suggest you try and find a similar caliber university in America (not the Ivy's but certainly the NYU's, USC's and the like) to do your undergraduate considering you should be eligible for some form of financial aid (which you likely will not get here) and apply once you have graduated. Like I say, if you are confident you could get into Bristol, a very good university, you can more than likely get into a very well respected American university which I promise Oxbridge will have heard of..
I know of a friend who attended Aberystwyth university (which I can guarantee you, or most of the people on here have never heard of) for their BA and got into Oxford to do their Masters and PhD in chemistry.....so suffice to say it is possible to attend Oxford, provided you excel in your undergraduate, no matter where that happens to be.