Hi ListenerFriendly,
My advice to you is: don't count on being able to get a UK job.
I'm a Brit who lived for almost a decade in the US, finished high school there, got my BA, my Masters, and worked as much as allowed on my student visa. I wasn't able to transition my experience into a long-term job due to visa restrictions, so I'm now back in the UK, not by choice (although obviously both places have pros and cons). Like you, I look great on paper and could do the jobs as well as any local candidate. But I found that if you don't have work authorisation and the employer won't sponsor you (most won't, as you say), you are SOL. Even if you want to work for the employer and they want to hire you, the visa is a huge barrier. For me, it was insurmountable. If you can overcome it, more power to you, but I would make some contingency plans as well.
Keep networking in the US and laying groundwork for a possible return. You'll miss out on a lot of US-based opportunities, so make sure you don't lose any ground on peers. Think of this as the opportunity cost of choosing to study internationally. You may not get the "halo effect" of your university's good reputation, and your networking opportunities may also be limited, so you have to work around that. If you're determined to work in the UK, you may have to compromise on industry/locations/pay/working conditions. If you want a job suited to your qualifications, you may be better off in the US. Sad but true.
One of my biggest challenges returning to the UK is that UK employers have no idea about my US university (which has a good reputation in the US though is not Ivy League), nor do they understand a GPA (mine is good), and even contacting foreign references is daunting for many of them. I imagine it would be similar in reverse (e.g., a US employer wouldn't have a clue that a 1st is better than a 2.1). I have a whole packet that I submit on request to employers explaining a standard US education, including how to interpret my SAT/AP scores, GPAs, transcripts, etc. I suggest you put something similar together that plays to your strengths. If the employer doesn't understand your background, you'll get overlooked, and I had not expected this to be as prevalent as it is. We study internationally, and we may one day work internationally, but for new grads, hiring is still very localised.
Sorry if this sounds negative. I wish someone had told me these realities when I was in your situation, so I am saying it to help you. I am sure you're awesome, but the market is just horrible right now even without all the extra visa hassles internationals face, and that employers don't want to know about. I'm not saying don't try, but be prepared for it not to work, and try and have a back-up plan for US jobs too.
Good luck.