No, it's been shifting more _liberal_, which is different. The effects of British and American McCarthyism still put Marxist (and also any associated leftist) thought very much out of the mainstream.
You can easily observe this on politicalcompass.org
They analyze voting records, speeches, etc. and use an objective test to score each party leader on where they stand on certain policies.
Here is a general yardstick for you, you might not agree with it, but this is the scale that I tend to use: https://www.politicalcompass.org/images/axeswithnames.gif
As you can see for America:
2004: The candidates are evenly spread around the map (https://www.politicalcompass.org/uselection2004)
2016: Almost all of the candidates are on the right-hand side of the map (https://www.politicalcompass.org/uselection2016)
One of the problems in US discussions that I've noticed as someone from Europe, is that Americans tend to talk about "leftists" and "rightists" to mean people who are respectively "less authoritarian" and "more authoritarian", regardless on where those people actually stand on matters relating to the left/right divide. I would say this is one of the biggest problems in US politics, as it causes the most misunderstandings when two people enter into a discussion.
As for the UK, you can see here that historically there is and always has been a right-wing bias, which fits with the (more or less) three party system that we have: https://i.imgur.com/yI0QXpL.png