Did the Nazi's ever invade England?

Yes, they invaded and occupied the channel islands.

-----

Actually, no. They did invade the channel islands, but the channel islands are not part of England or even of the UK. They are a crown dependency. They are remnants of the old Duchy of Normandy, which never reverted to France, and are separate. It is for this reason that one of the titles of Queen Elizabeth II is Duke of Normandy (note: properly "duke," not "duchess," in this case).