I'm a little confused about your idea that there are parts of Scotland that are "Anglicised" or subject to "English influence through families".
I would suggest that you start visiting different parts of the U.K. In your next holidays because you seem to have e done very odd ideas.
There are all sorts of names in my children's school. From all over the world.
A large number of the names you no doubt consider "English" are biblical names so have been used all over the world for a long time. So for example;
Elizabeth
Sarah
Ruth
Hannah
John
Matthew
Andrew
James
John
Thomas
Daniel etc etc
People in Scotland might also use some Gaelic versions of these names eg
Ian for John
Hamish for James
Elspeth for Elizabeth
But John, James and Elizabeth have also be standard names in Scotland for hundreds and hundreds of years.
Not all of Scotland was ever Garlic speaking btw large parts of the country spoke Scots instead.
Yes there are lots of S Irish names you might not know how to pronounce, but I'm sure there are also French names you don't know how to pronounce. Spend a bit of time in the country concerned and you'll learn.