moved back or has moved back?

She used to study English with her private tutor everyday but he moved back to England last month.

both are quite acceptable.

If you specify a point in time (last month) you need to use 'moved back', otherwise 'has moved back' is also okay.

She used to study English with her private tutor every day but he moved back to England last month.
or
She used to study English with her private tutor every day but he (has) moved back to England .

You need to use 'every day' not 'everyday'.
http://www.elearnenglishlanguage.com/blog/english-mistakes/everyday-vs-every-day/

So far so good

A., however, Bear, seems a little clumsy, dunno why, maybe just me

Sorry, I don't agree with that.

...but he has moved back to England.
or
...but he moved back to England last month.

sound better to me.

Yes..but that's am opinion. The point is that both are grammatical.

"has moved" reinforces the fact that it (his moving) is IN FACT, a done deal. It's done been done.

Personally, I like it when I say things in such a way to make them sound "double strong." Like when I say, for example: "And that ain't no lie!!"
1. It aint a lie, and, not only that, but also, too...
2. It is "'no lie."

I think my carp is more a matter of style, but we need more context or more examples to bring out the point.
I will happily concede that both are grammatical, more or less, but one is very clunky.

"last month" puts this into a simple past tense.