In General Europe is still save as usual.
I was few days after the start of the war in Poland (near the border) and yes it was quite busy due the refugee´s but Poland raised pretty fast a system to transport and accomodate them. Currently the refugees will be have to go through a “Registration” and then will be guided quickly to trains westwards Because of that the trains from Poland to Berlin are currently extra long and still quite full (at the beginning crowded). The refugees´s are seperated from the tourist accomodation as they usually will be guided more and more west and have so called “Tranist Centres or Welcome Centres). For example next to the Main station in Berlin they raised a tent with informations and they reopened the former Airport Berlin Tegel as a “Hub” with sleeping places and a refugee managment. (Buses bring them to their destinations or to the next trainstation)
few railcompanies even run special trains for them. So the system in Europe gets back from “crowded” more and more to business as usual.
The trains from Budapest & Warsaw to the west are currently still busier than usual but it works.
For me it´s like 2015 but ofcourse now from a other direction.
The situation in Ukraine doesn’t really have a bearing on travel in Europe in general. The war is restricted in its spread by security alliances (NATO on one hand, and whatever Russia and Belarus have agreed between themselves on the other) which would - in the unlikely event of direct conflict breaking out somewhere like Poland or the Baltic states - instantly make everywhere else in western Europe just as dangerous to be in, including the UK.
You are right in thinking that there is a concerted effort being made to put up refugees, but if you’re concerned about hostels filling up you can always book early. I’m not sure what proportion are being housed in hostels, as the thought hadn’t really occurred to me. I’ve so far booked two hostels for my own trip this summer, so you are now within the usual booking window.
However, it’s only really crucial to ensure you successfully obtain a hostel booking if you’re travelling alone; when sharing a basic hotel room or a short-term apartment let with one or two other people you can often get similar prices per person if you’re organised and flexible with your bookings.