I attend a large regional eye hospital for my contact lenses.
I think that the main advantage of attending a large eye hospital is that they have access to a large range of specialist contact lenses and the expertise to fit them. The other advantage is that the cost of the contact lenses is at the NHS price. The main disadvantages are travelling to a regional centre, availability of appointments and that you main not see the same optometrist.
A local hospital may fit contact lenses but may not have the same expertise and range of contact lenses, or they may refer you to a community optometrist.
A community optometrist may fit specialist lenses, but may only have a few KC patients. If you were referred by a hospital you would probably be paying the NHS rate for the lenses, if you went privately you would pay the full cost. The main advantage would be that they would be local to you, you would see the same person, and you would be able to get an appointment easily. But if they only fit a few patients they may not have lenses to suit you.
Alternatively there are optometrists who love a challenge and specialise in fitting complex cases. These optometrists are likely to have access to a large range of the latest lenses. However they could be very expensive - but if they can find a lens that works for you it could be worth considering.
It might be worth contacting a few optometrists to find out whether they fit complex cases and what their charges are.
There is a list of KC friendly optometrists on the website:
http://www.keratoconus-group.org.uk/optoms/index.phpLia