I used to work in a university mortuary and routinely handled the bodies of people who donated their bodies to science so feel free to PM me if you have any questions about the process.
As most people have said, you will have to contact your local medical school and they will arrange bequethal paperwork to be sent to your father. In the UK, all accepting organisations need to provide informed consent. All the information will be in the pack and he will need to sign them himself, it cannot be done by another family member.
In many cases, we have had people sign up to donate their bodies and the family members decide after they have passed that they don't want to go through with it and that's perfectly ok, if you feel that strongly about not doing it. There is no obligation to continue with the process. When the time comes, you can simply phone the organisation and if you want to continue, they'll guide you through the entire process or you phone up, say they've passed but you don't want to continue with the process and that is the end of it.
There may also be the possibility that his body won't be accepted and this could be due to a number of factors and in this case, you will have to make your own funeral arrangement.
The universities at I have worked at, and I don't know any who don't, provided a funeral service each year for the bodies who have been donated. In most cases, a body can be retained by the organisation for 3 years so it may take the full 3 years but you will be invited to a funeral service. Where I worked this was held in the university chapel where the minister, lecturers from the university and students all gave accounts of how much donations mean to them.
I don't know any in the UK who return the body for burial, in almost all cases I've know, the body will be cremated by the organisation and you have the option to collect the ashes or have them scattered by the organisation (my university had its own memorial garden). The cost of this is covered by the university. As are any transport costs for collecting the body (unless the body is coming a significant distance then there may be extra factors).
There are a lot of articles out there that are scary but please be aware a lot of that is out with the UK. The UK have a very strict set of legislation, rules and criteria to ensure things like that don't happen. The gift of someone donating their body, as many medical professional will tell you, and many people have said here, is so incredibly valuable to the teaching of the future of medicine and is treated with the utmost respect and dignity. I have first hand seen the number of young medical students, starting their degrees and their first steps in their careers in the dissecting room and the experiences and knowledge they receive from physical cadavers is so much more than they could ever get from books and apps and digital learning.
I appreciate how hard it can be for families knowing they might not have a 'traditional' funeral service. I had a relation die who had chosen to donate their body to science and while I was arranging transport, was told the family had changed their mind so please don't think that never happens and you'd be the only one because you're not.
Hope this helps.