Upvote:2
Yes, as a US citizen you are allowed to travel to the US, as can be found in other current answers on this topic.
As a (presumably) UK resident you are also allowed back into the UK, but have to observe a 14 day period of self isolation. See https://www.gov.uk/uk-border-control for details.
For travel to the rest of Europe, you can use the Re-open EU page which shows restrictions for the individual countries - countries not in the EU of course have their own rules. https://reopen.europa.eu/
For most Schengen countries, third country nationals are not yet allowed into the country without restrictions, as having to self quarantine or testing etc, and some countries don't allow third country nationals entry at all.