Here's a quick update (after some more testing). The situation is really random...
1. The UNA.io website works perfectly. It saves my full address and when I search by street, city or country - I appear in the search results. Even when I set Kyiv as my city.
2. Our installation doesn't work properly (as described above).  I can get the city-based search to work, but only for Budapest (or any other city) but not for Kyiv.
3. The UNA demo site (online.me) has similar problems, but they are totally random...
If I add a full address to my profile, it removes the street and saves the city /country.
If I add Budapest as my city, it removes the city name and saves just the country (Hungary)
If I add Kyiv as my city... then it saves the city name and the country
If I choose Oxford University as my address, it saves just Oxford, UK....  
It also behaves differently depending on how the address is searched (slect with the mouse and it works, use the keyboard keys and it doesnt work)
I'll try to upload a video/screenshot tomorrow, but I guess you can test online.me 
I'm afraid I'm new to UNA, and have no idea how the location system is working, but this is the situation we have.
If I could be certain that it would work on a UNA/Boonex hosted version of the site, I could be persuaded to upgrade, but the fact that its not working on online.me worries me.
I hope there's a relatively easy solution :)