Budapest - Vienna has optional reservations, like the vast majority of European trains. You are always allowed to board but may have to swap seats along the way.

You can get one for 3€ on ÖBB (free with a 1st class pass), including selection with a seat map. Search reservations only, it is the same process whether you have a pass or a regular ticket (which do not include reservations).

They may not be available right now because there are some last-minute engineering works on the railway. Check closer to travel date.

More info about reservations : https://www.seat61.com/interrail-and-eurail-reservations.htm

They're always cheaper when going through the company. They're even free in the UK (do not pay anything for them !).

The Eurostar needs to be booked pretty soon (popular route and passholder quota). Check availability here : https://www.b-europe.com/EN/Booking/Pass#TravelWish

Questions welcome. :)

You don't need reservations for most of that journey. Mostly you can just turn up and find an empty seat

Reservations are compulsory for:

Regiojet trains Budapest- Vienna-Prague (but Railjet trains on the same route don't need reservations)

Sleeper train from Berlin-Brussels, if you choose to use it.

Eurostar from Amsterdam or Cologne-Brussels if you go that way. (But the Intercity and ICE trains cover the same route and don't require reservation.)

Eurostar Brussels-London.

On all other parts of your route, reservations are optional. They are free or cheap if you make the reservation in the right place. In any case, there's no need to do it now unless you really know you will commit to that train. Better to stay flexible.

Agree with @thibcabe . The key reservation you need to make here is Brussels-London. Do that as soon as you can. 

I would also book any other Eurostar or sleeper trains you intend to take. 

Anything else is no rush.