How likely is it that a UK or EU PhD in STEM has to be self funded?
Here's the thing: Doing a PhD - perhaps not in every single discipline, but certainly almost always in STEM - is a form of employment. You would be a junior researcher, with a measure of independence and a measure of supervision, and with bosses, a department and a university to answer to. You'll need to be on campus most of the week (albeit with the possibility of remote work depending on specifics); you may be required to teach if the university needs you to; you will be subject to all rules and regulations the senior academic staff is subject to, etc.

In those EU states where this is recognized, an "self-funded STEM PhD" is like "volunteer work"; universities can't rely on its existence, and if they want PhD's they have to shell out the money to pay them their wages. Not that you make a lot of money, but it's something. There are sometimes collective labor agreements which govern PhD employment.

In other states - including, AFAICR, the UK - PhD candidates are not recognized as employees. In those states, self-funded PhD are more of a recognized option, but still obviously very rare, as few people can afford to do full-time volunteer work for years.

And thus...


  How likely is it for a physics phd to be self funded?


Very-to-extremely unlikely.


  Do most PHDs get funded by the university or other third party funding?


This depends on the state in the EU, the university and even the individual researcher, there is no general answer. Also, it sometimes happens that you're funded by the university for some parts your PhD and by a third-party source for other parts - or even that, at the same time, both the university and one (or more) third parties fund your employment (or "stipend"). Finally, third-party funding sources sometime get pooled at the departmental or university level, so that "university-funded" can mean money from different sources.


  £20,000 a year for 4 years is not an option for me if I have to pay these fees myself.


Indeed it is not, and please don't try to do this kind of volunteer work, because you'd be hurting your fellow PhDs by legitimizing such a practice.