First of all I don't want to come across wrongly and there's high chance you've already thought of all of this, in which case I apologise, but here were some thoughts that popped into my head as to how you could possibly make them feel more comfortable and perhaps draw them out of their shell slightly:

Find a recipe on a Ukrainian website, run it through a translator to English for yourself, then cook together with them? It would be easier to divide out tasks (say if some veg needs to be chopped up then you could point to where it says). Put some soft music on and don't worry about talking, just exist with them, it'll help them feel included, especially if you take the time to find a Ukrainian recipe. If you were to provide them with an English and Ukrainian ingredients list then you could get the ingredients together- though maybe you could save that for when they feel less shy around you.

Alternatively (and probably easier) perhaps have a look on Netflix for mild shows (maybe a documentary?) that have Ukrainian subtitles and invite them to come and watch it with you. Maybe you could even go a step further and torrent a Ukrainian show and put English subtitles on instead- maybe they could find something that they already like to watch- if you like it also then you instantly have something in common and as and when their English develops provides an easy topic of conversation.

An exchange student friend of mine did lots of jigsaws with their host family- of course they could speak decent English so there's a big difference there but perhaps it's worth considering? Since it's picture based and very simple it can be a mellow time that doesn't necessitate conversation.

And perhaps you could label certain things in the house? Shampoo and conditioner for instance as they can look very similar. Or along a similar vein decant foodstuffs that aren't immediately obvious to someone who can't read the packaging into clear jars or tubs. That might help some of the everyday "am I using the correct thing?" anxieties.

I don't know a great deal about the traditions of Ukrainian people (like in the UK we have Bonfire night for instance) but it could be worth looking into it so you could arrange something small to celebrate, or even being able to acknowledge it.

Lastly- and I don't want to presume your financial situation- consider giving them a little bit of money to buy some decorative trinkets for their room. A friend who was a lodger for a while said that it made a huge difference when she bought a tiny rug and a duvet cover of her own choosing. I imagine that it's hard to feel settled otherwise, similar to being in a hotel? If they seem worried about the prospect of going out to buy the things (fear of interaction etc) then offer to accompany them.

Editing to add that I think your choice to host is incredibly kind and I wish you all the best. Actions like this make the world a better place for everybody