Do hornets return to the same nest after winter each year?
If fail to get rid of the hornets this fall, will they return to the same nest next year (or overwinter and continue to live there)?


No. The nest, likely made of wood pulp (and maybe mud), will/should degenerate to unusable over the winter, though the bulk could remain for a couple of years. All wasps die, except for the new queens (fertile, mated females) who overwinter in sheltered crevices somewhere.

In spring, it is a single wasp that starts a hive: the new queen. She builds a new, small hive in which to lay eggs, which is added to initially by the queen and then by workers who have hatched.

While she will not return to an old hive, the conditions which appeared advantageous to one queen (shelter from rain, intense sunlight, winds, etc.) will also seem advantageous to other queens, which is why nests appear in the same places year after year.

Edited to add: It appears most likely that the queens that overwinter are "new queens", that is, mated female offspring of the queen. H/T @Brian Krause.