Member states still have to temper division within their own countries and that includes countries where the right have gained power. Look at France for example, Le Pen has gained a lot of traction and votes but France is divided just like the UK.
The right in France has united behind Le Pen and that has pushed her over the line but only in terms of seats in the French electoral system. 32% voted for Le Pen but remember that 50% voted for two parties that oppose Le Pen. If those two parties combine then she will likely have the most seats but get zero power.
With tomorrows expected result you could argue that the left is on the rise in the UK but it's not true. The reverse of France will happen tomorrow, Labour will win because the right has divided between the Tories and Reform. If Reform voters for example instead voted Tory today then Labour probably would not win tomorrow.
All of this rise and fall is just a product of electoral systems and not the rise of any particular ideology.
Brexit is a divided issue where we'd almost certainly need a general election which votes in a party that wants to rejoin. I just can't see that happening because certainly Labour cannot win on rejoin (they were obliterated on this in 2019) and there is no other party who will propose it.