Unfortunately you will be unlikely to find help in this regard.
I've never seen this kind of contract in the UK, unfortunately. Vets aren't paid much in the UK (and exotic/wildlife vets are paid even more poorly), so even if you found someone willing to fund your studies, you would probably have to work with them for decades in order to fulfill the contract and pay back what you owe.
Have you previously done work experience in vet clinics? Your future plans sound great! But there's a lot more to being a vet than you may realise. I strongly recommend doing some work experience in a clinic before you progress with your plans. We're having a huge mental health and retention crisis in vet med right now, because many people do not fully understand the role of a vet. You will also need some recent clinic experience in order to apply for vet school.
Wildlife vet is a very competitive job; many students wish to do this. You need to get on board with the idea of doing domestic animal medicine, as this proves to the vet schools you are serious about the career. You would need to study domestic animal medicine for 4+ years at university, and practice domestic animal medicine for several years in your first job/s before you will have the experience required to go for an exotics or wildlife role/internship/residency etc. It's best you learn to be passionate about this as well!
Also, the role you've described here makes me wonder if you might be better off going into a conservation role. Are you passionate about medicine itself? Because whether you're working on kittens or lions, your role is going to consist of similar things: monitoring anaesthesia, calculating drug dosages, running diagnostics, administering medications, euthanasia etc.
If you really want to be able to become a vet, you will probably need to give up your dream of going to Edinburgh (unless you have family members who can fund this). You'll be unlikely to crowdsource this amount of money. The English vet schools tend to be a lot cheaper for graduates, and all vet schools in the UK are of world-class standing, so Edinburgh is not necessarily better - other vet schools also have great wildlife departments? Not to mention that vet school is incredibly competitive (as you already know from not getting previously) and graduate veterinary medicine is widely regarded to be even more competitive. If you only applied to Edinburgh, the odds would be stacked against you. It's best to apply to multiple schools and not put all your eggs in one basket.
For the UK fees of £9250 p/a, self-funding is very much doable. Many of my graduate friends on the course manage to work enough during the summer to fund their tuition throughout the year and in addition, you are still entitled to maintenance loan. If you can settle for vet schools other than Edi, then you can much more easily afford to study veterinary medicine.
I hope this helps you somewhat. Good luck on your journey.
Edit: I see you've already applied. Ignore the bit about work experience in that case!