Are you really claiming that French people are fine with recent German immigrants, and that Germans don't mind the Italians, and that Welsh people are ok with the English (and vice versa).

In my experience, mostly yeah, at least among younger, more educated people. Among older people, there may still be quite a bit of resentment though. That's kind of the whole selling point of the EU.

The reason this does not come up much is that there is not that much migration between those countries.

That depends on which countries you are asking about. There's quite a bit of economic migration from Italy to Germany/Austria, for example and yet there is little anti-Italian sentiment (at least about migration).