How much does an undergraduate degree matter for an academic career?

I'm a PhD student, in medical physics at a very good collegiate university in the UK. I did a four-year undergraduate masters' (in physics), and I'm just finishing a four year PhD. I'm now at the awkward stage of applying for grants, junior research fellowships and postdoctoral fellowships -- and I'm experiencing an awful lot of rejection letters, often after being shortlisted and interviewed.

My question is this: how much does my undergraduate degree actually matter at this point in my career? I wrote two articles (on computational biology) as an undergraduate that were published in good (impact factor ~5) journals, and by now my publications list has eight items on it (excluding conference proceedings), including a PNAAS article (albeit not first author) and more than one in the main journal in my field. I've won prizes, and got a teaching qualification. Yet I still keep being rejected for positions that are 'appropriate' for me to apply for.

When I speak to older (successful) colleagues about their experiences, they often drop things like "Of course, coming first in the year at [Cambridge/Oxford] helped me get my Junior Research Fellowship, and even the Tutorial Fellowship later" into conversation, and the vast majority have a high first. I didn't do fantastically in my undergrad degree -- I narrowly missed out on a first class degree, and got a high upper second (69.96%), largely due to one bad exam. I really can't help but think that the reason I'm finding it so hard to get funding is because I didn't come first in my year -- but I'm up against people who presumably have successful publication histories and did.

If there are a large group of equivalent, good, candidates for one position, do funding bodies and interviewing committees look at what's different between everyone? Is the fact that I'm objectively a second-class physicist holding me back? If so, what can I do about it? Or is it the case that these funding bodies do some sort of crazy weighted sum, whereby one-tenth-of-a-nature-paper is equivalent to being first-in-year?

I realise that I should feel pleased to be shortlisted where -- to give an example -- 283 people apply for one position, and I was in the final six. Yet 'feeling pleased' won't pay the rent next year, and I'm really starting to despair. Should I accept that this limitation is always going to hold me back in my chosen career path, and therefore just go and change it?

Aside: I'm also concerned that, being an interdisciplinary person -- an MRI physicist -- I'm going to come across as being "too medical" for a physics position, and "too physical" for a post in a biochemistry department. In practice, my research ranges from Schrödinger equations to talking to cardiologists, and I believe that either location would be appropriate. This, however, is a whole other kettle of fish!