First of all, you have a legitimate excuse for not being to get the GCSEs etc. so don't beat your self up about it.
Age shouldn't put you off, nor the fact that all your friends are at uni. Certainly, many people end up going to uni later! At Liverpool, the oldest student we have currently is in his 50s. Of course that's unusual, but we have plenty of people starting vet school in their 20s (a lot who did a degree before hand or worked and then decided that, actually, they wanted to become a vet). You don't need to be embarrassed at all, even if you graduate in your 30s then you have 30-40+ years ahead of you to work as a vet. Being older might also be an advantage - clients will trust you more (they will asume experience if you look older) and you will probably have a more mature attitude to everything that a lot of 18 year olds lack. You will probably find that a lot of your friends won't end up using their degrees - for some it might be because they didn't really know what they wanted to do at 17/18. So it's probably better that you know what you want to do, then get a degree in a course that you lost interest in.
I don't mean to pry but when you say you are almost fully recovered, you need to be aware that vet school is stressful and demanding mentally and somewhat physically to. Plenty of us cope with various medical/mental conditions but it's not be taken lightly before you apply.
In your situation, I would email or call all the vet schools. Tell them you situation and with what qualifications you will need to be accepted under your circumstances so you know what you need and where you stand. Some have foundation courses (year zero) courses which you may be eligible for. Each vet school might give you a different response. There is no easy way in though - so you might end up having to do at least some of the qualifications that you missed out on. If you really want to be a vet, then the time and finciancial expense will be worth it. I've had friends who went back into education as adults (not just for vet school mind you) for either GCSEs/Alevels. You should be able to find evening classes or adult classes. As you have no GCSE qualifications - is there any government funding available for you perhaps??