Overall as a university across all subjects on average, Edinburgh is only slightly better than Manchester, according to Edurank (Edinburgh 4th, Manchester 6th, with Imperial inbetween). Topuniversities.com puts Edinburgh 5th and Manchester 6th.
The Complete University Guide puts average research quality across all subjects on average as 87% at Manchester, 86% at York, 85% at Edinburgh, Lancaster and Leeds both on 84%. For this subject, 90% at Edinburgh, 89% at Manchester, 83% at Lancaster, and 82% at Leeds. University of York isn't stated there.
But research quality gives no necessary indication about the facilities (and the amount of research if that matters to you). Eduranks puts Manchester at 4th, Leeds at 8th, Edinburgh at 14th, York at 26th, Lancaster at 62nd, for Biomedical Engineering.
For student satisfaction across all subjects on average, Edinburgh and Manchester are nearly in the pits, on 74% each (so are Leeds, Cardiff, KCL and QMUL), according to The Complete University Guide. I am speculating but the high percentage of international students at Edinburgh, KCL, Manchester, and Leeds must sometimes make it harder for the teachers and some of those students alike, so low average student satisfaction does not necessarily represent the satisfaction rate of their UK born students (or, who knows, perhaps it does, if they'd rather be with more UK born students).
York is on 77% and Lancaster 79%. Those last 2 are 60s campus universities, which can be very Marmite. Even those who like the campuses as a working environment (I like some 60s buildings - to visit) wouldn't necessarily want to live on one - it's all a bit artificial feeling and 'samey' to me, and the lack of old buildings (apart from Heslington Hall at York, which was presumably too good to give the students and is used for admin. The Archeology Dept, miles away in the city centre, is old) can make it liminal space-like, although the lake at York and the UFO Central Hall are a nice touch.
For Biomedical Sciences, student satisfaction is 84% at Lancaster, 76% at Edinburgh, 73% at Leeds, and 71% at Manchester. University of York isn't on there.
Graduate prospects on average for the subject (this may just be in the opinion of the students themselves) are 92% at Lancaster, 88% at Edinburgh, 86% at Manchester, and 73% at Leeds. University of York isn't on there.
Entry grades actually achieved for this subject are 92% at Edinburgh (remember, at Scottish universities, the percentage is always more because Scottish students take more Highers than the number of A Levels that the rest of the UK are encouraged to take), 79% at Manchester, 76% at Lancaster, and 74% at Leeds.
Bearing in mind that you care about the general prestige of the university compared to others too (to save time, I haven't put any London universities (n.b. or any outside of England, or further South than Bristol) because all your choices are in the North of England anyway):
Another university with all round pretty good looking stats for this (although it is a 60s university) is Bath University. For this subject, research quality of 80%, student satisfaction of 80%, graduate prospects of 89%, entry standards of 85%. Edurank ranking for this subject: 24th.
And Bristol. For this subject, research 88%, satisfaction 75% (still better than 3 of your choices), prospects 84%, entry 80%. Edurank: 12th.
And Sheffield. For this subject, research 90%, satisfaction 78%, prospects 87%, entry 75%. Edurank: 7th.
And Nottingham. For this subject, research 81%, satisfaction 75%, prospects 79%, entry 67%. Edurank: 6th.
And Birmingham. For this subject, research 85%, satisfaction 73%, prospects 87%, entry 73%. (none of these stats are better than Sheffield's though). Edurank: 15th.
And Warwick (60s campus). For this subject, research 81%, satisfaction 78%, prospects 91%, entry 72%. Claimed prospects for it there are impressive, considering entry standards are considerably lower, although research quality isn't as stunning for it. Edurank: 32nd.
And, because its research quality is so high for this subject, and because I went there for English, I also want to mention Leicester. For this subject, research 89%, satisfaction 74%, prospects 75%, entry 65%. Edurank: 37th.