British nationals can certainly travel to the European Union, even without being required to quarantine. But if a national of a country that is not on the list of allowed third countries travelers have been in Britain long-term under a long-duration tourist visa or visa-exempt passport, but without a residence permit, will they be denied entry to the EU? Or will they be admitted? Would there be certain documents that they would be expected to provide?
That depends on the EU country. As you can see on https://reopen.europa.eu/en and https://www.iatatravelcentre.com/international-travel-document-news/1580226297.htm, each EU country has a different policy. For example, from my understanding of the current policies, as a someone who has been in Britain long-term under a long-duration tourist visa or visa-exempt passport, but without a residency permit in the UK, can travel to Croatia as a tourist (see IATA "passengers traveling as tourists with a confirmation of accommodation booking") and France regardless of your intent (see https://reopen.europa.eu/en/map/FRA: "voluntary quarantine for travellers from the UK."; no need to be a tourist).
Per TIMATIC, you can enter Spain in the following cases:
- If arriving from EU/Schengen/UK
- If residing in EU/Schengen/UK, Andorra, Australia, Canada, Georgia, Japan, Monaco, Montenegro, New Zealand, Rwanda, San Marino, Serbia, Thailand, Tunisia, Uruguay or the Vatican
- If holding a Schengen D visa
- If eligible to hold an Article 10/20 family member residence card