I have heard that roaming charges have been eliminated between EU countries when you have an EU phone number. Has that started and does it include UK numbers?
Yes, for as long as the UK remains part of the EU, EU consumer protection rules apply.
What happens after that (post 2019) depends on the results of the negotiations. Anybody's guess.
Some UK companies have extended no roaming charges to other territories outside of the EU such as Turkey and Jersey.