We have a massive oak tree in the front yard, and the ivy (English) that
covers the ground under it has grown up onto the tree. We recently bought
the house, and I'd guess this has been going on for 10 years--the ivy is at
least 25 feet up the trunk, and well-established.

I want to remove the ivy from the tree, so about 2 months ago I cut all of
the vines on the base of the tree--basically created a "clear cut" circle
about 1 foot up the trunk. cut off from the ground I expected the ivy on
the tree to cooperate by dying out, and then I'd pull the dead vines off of
the tree.

Right.

The ivy on the tree seems to be as healthy as ever. Is that a surprise?
Can it live without access to the ground? Does that mean it's getting
water/nutrients directly from the tree?

How should I go about removing/killing the ivy from the tree? I'm not
opposed to applying Roundup or something similar, if I can do so w/o
harming the tree.

Any recommendations would be appreciated.