I’m assuming you used the UN News global ranking in physics, where Manchester is 43 and RPI, 523! I don’t know how accurate that is, I grew up in upstate NY and RPI had a pretty good reputation for its physics dept, but that was in the 80s, so things may have changed. If those are even somewhat accurate, that’s a huge difference and I think may put Manchester back in play as the choice. Also don’t worry about the MPhys vs BS, most, if not all, PHd programs in the US will give you a masters as well if you want it, along with your PHd. A friend was applying for physics PHd at Cornell and he said they don’t have a masters, you go straight from BS to PHd. Good luck!
I don’t know anything about either of those schools, so I can’t comment on the level of prestige. However, as someone outside of the physics realm, I don’t think the University of Manchester is that well-known in the US (not like Oxford or something).
Are you 100% sure you’re going to do a PhD? Would you still think it’s worth it if you change majors? What if you change to something that requires a Master’s degree instead? The PhD factor shouldn’t change how much debt you’re willing to take on, IMO – in fact, you’d likely make less from your PhD stipend than you would working with a bachelor’s degree, so you’d have to wait until you get your PhD to start paying off your debt.
And isn’t the conventional wisdom that prestige matters LESS for undergraduate programs if you’re planning on graduate school? Employers won’t care about how prestigious of a college you went to as an undergrad, just where you went for your PhD…maybe focus on prestige then. 60K is a lot, although it’s doable.
“You could also split the difference. Go to RPI undergrad and apply to UofM for the Phd. Makes it free and if you think UofM is best, then you’ll gain in both respects.”
That is not likely to be a feasible option. PhDs are not funded by default in the UK, you need to get a scholarship or research grant to cover the costs or else you pay yourself. Most grant funding that the universities have is for UK/EU students not internationals - so you may end up limited to seeking ultra-competitive scholarships (eg Marshall or Fulbright scholarships).
That’s because you just do research and finish in 3 years, you aren’t TAing and taking 6 years as is more typical in the US.
@theloniusmonk I didn’t look at USNWR - but if you look at QS you’ll find RPI in the 150-200 tier 
But more importantly, those rankings don’t speak to the undergraduate physics program in particular- and those are (comparatively) flat, especially in the first 2/3 years. My physics kid, and her friends in Ireland, Scotland, and Belgium compared their undergrad physics degrees (they are now all in grad programs), and the differences between what the European kids did and what my US-based LAC kid did physics-wise was considerably less than any of us imagined it would be (especially as they did physics 100% of the time for 3 years, where mine did it 60% of the time for 4).
To be clear, UM is a great university, and in a world where the OP could afford to go without taking on the debt of a new sports car I would be all for it. But the specific question was, how much is the extra prestige worth? and imo, it’s not worth the debt. Prestige of undergraduate school is not a big factor in physics grad school admissions, and the OP can absolutely get into any physics grad program from RPI.
Does the prestige comparison matter at this point? The OP’s parents are financially forcing him/her to attend RPI (despite RPI costing them more). It does look like RPI graduates in physics do commonly go on to graduate school, but the OP may want to contact the department directly to find out how many years and total graduates the listed examples of prestigious PhD program matriculations come from.
https://science.rpi.edu/physics/programs/undergrad/bs-physics
I am not a fan of comparing the “prestige” of different universities. Instead please focus on the resources and opportunities available to undergraduates.
I am a physics professor who is familiar with the physics department at RPI. Robert Resnick (of Halliday & Resnick fame) and his successors developed a strong introductory physics curriculum there. The department offers all of the courses and research opportunities necessary to prepare for physics graduate school. RPI is well known to US graduate programs and their alumni do fine in graduate school.
I doubt that the University of Manchester would give you a leg up over RPI in US graduate program admissions. As others have noted grades and GRE scores count, but research experience and recommendations from professors are the most important factors. RPI can offer you those things.
As has been mentioned, many PhD programs (unlike Med School or Law School) are funded programs. Middle brother did one of these (neuro something or other) and was admitted into 3 or 4 top programs, all of them being fully funded.
So your issue is really more about undergrad costs.