Hey,
Unfortunately I can't be of much help on the Durham front, and only very little on Warwick, but I can give my two cents on Birmingham. (I applied for Biomedical Science in December for 2015 entry)
Basically, Warwick was on my shortlist of Universities to apply to, however in the end it did not reach the final 5 as, even though the University itself is excellent and has a stellar reputation, I just didn't feel the course was that great. At least not for me - of course, it's all down to personal preference which courses look good or not. I wasn't keen on the modules, layout, options etc. Doing my research on the department/teaching, I generally just found it just wasn't that great in comparison to the other Universities that made my final 5. I felt as though they had stupidly high entry standards simply due to the reputation of the Uni itself, which did not reflect the quality of the course and teaching.
Now, Birmingham - In the end I applied to Birmingham, Surrey, King's, Leicester and UEA. I've received a rejection from King's, am still waiting on a reply from Surrey, however I have recieved 3 Unconditional offers from Birmingham, Leicester and UEA.
In my opinion (and again, it really is all about opinion and personal preference when it comes to Universities so close together in reputation, rankings etc), Birmingham is the better of the Universities, and after I have visited the applicant day I plan to confirm it as the Uni I will be going to.
Reputation/Facilities
For one, Birmingham has an excellent reputation not just as a University in itself, but in all things Medicine and, consequently, Biomedical Science. They have Class-A Cancer Research Facilities among other stellar Medical-related facilities and opportunities.
The Course
But aside from that - the course. Now, the course isn't as bespoke as I would have liked it - it's pretty rigid in terms of you not being able to choose optional modules until Year 3, and even then you can only pick 2. 2 optional modules in 3 years is tough if you really want to tailor a course to yourself. (However, I have found writing somewhere that has alluded to being able to choose Modules in Year 2 as well - not sure if this is an error, or perhaps something to be implemented in the future, but it would certainly be nice). This was not a deal-breaker for me though, as the modules you do get to choose in Year 3 are, by far, the most interesting modules I have encountered so far. Things like Neurotrauma, Brain degeneration and Regeneration, The Biology of Ageing, Neurobiological Mapping - of course, again this is my own personal interests in these fields coming through, but they have loads of ridiculously interesting modules for those more inclined towards other body systems or pharmacology etc. I'd also like to add that rigidity in courses is also probably a good thing - It makes sure you study what is essential to a Biomedical Science degree, to ensure you have the best knowledge and thus employability upon graduation. Courses like Warwick offer more module choices, but let you stray so far from the path that future employers are probably going to look at your degree and wonder how on earth you're supposed to go forward now you've completed your learnings of subjects that don't even resemble the basics of Biomedical Science. There's a reason the IBMS doesn't accredit many courses - because most courses don't offer enough subject material required to be officially recognized as a course in true Biomedical Science. Now, Birmingham is not accredited, nor is Warwick, but in my opinion Birmingham far more closely resembles what a course in Biomed should look like. It hasn't been butchered for the sake of freedom of choice.
ANYWAY, sorry for waffling on, I do hope I've helped in some way though. Again, I can't stress this enough - it's really down to personal choice. But when it came to choosing where I wanted to study, to me Birmingham did better than Warwick in every aspect of the Biomedical Science degree, and yet still had lower entry standards simply because they reflect the quality of the course itself, and not the overall reputation of the University.
TL;DR - Birmingham all the way.