The OP is traveling on an EU passport which makes immigration/passport control much quicker. I don't see an issue with this except that the gates may be far apart and require quite a bit of walking.

While UA is technically responsible for the rebooking if they deliver you late to FRA, LH in all practicality will be the one to rebook you. It's usually done automatically but if you miss the connection just head to the LH service center which are available in many spots and they'll get you on the next flight no issue.

There's not much to worry about except one hour being tight. Rebooking is easy and no issue (assuming there are two seats on the flight to put you on, of course).

-RM