It's going to depend on the outcome of the war, somewhat. But, generally, the ways that languages in contact influence each other depends on the nature and frequency of their interaction. It's also going to depend on how speakers of Ukrainian feel about the Russian language itself (in contrast to how they feel about the Russian government, army, etc). If they associate the Russian language with occupiers, it's very reasonable to expect them to strengthen their resistance to it.

A few examples of that happening historically:

1. after Iceland's independence from Denmark, there was a concerted effort to remove "Danish" features from the Icelandic language.
2. after the American Revolution, the same happened with American and British English. Webster, whose school grammar book became standard, intentionally made changes to help make American English distinct from British English: removing the u from words like colour/color; reversing the -re in words like theatre/theater; etc.
3. the same happened with English and French during the 100 Year wars. In fact, that's one of the main reasons English still exists. It had declined in use after the Norman Conquest, but was revived in the 1300s, for a number of reasons. Henry IV was actually the first King of England since the Conquest to speak English as his native language. Some of the earlier kings didn't speak English at all.

Without a negative response to the use of the Russian language, the two are somewhat related, and depending on the level of contact between them (which I suspect is high), there could be longer term influence. That influence usually operates at three basic levels: words, sounds, structure. (meaning, first we'd see shared word use and/or word borrowing; then sounds, then sentence structures/grammar like word order, methods of negation (how you say 'not'), etc.

That longer term response is both natural and a more likely reaction if say, Russia were to conquer Ukraine and make an effort to replace Ukrainian with Russian, or if the war is short lived, they resolve their problems, and the interactions between the two languages basically continue as before.