Reservations are optional in the UK, mandatory on Eurostar and night trains.
Edinburgh - London: reservations are optional but recommended. Available for free on the LNER website.
London - Amsterdam: I'll have a look but most trains are likely sold out. Eurostar is extremely popular around Christmas + there's a passholder quota. Some options:
Option 1: 23rd December Eurostar London St. Pancras - Amsterdam Centraal 06:16 - 11:15 1st class only (43€)
Option 2: 22nd December Eurostar London St. Pancras - Paris-Nord 16:31 - 19:48 1st class only (38€) or 18:01 - 21:18 2nd class (30€). Mandatory night in Paris (not ideal).
Option 3 (the best one, no doubt):
- GreaterAnglia London Liverpool Street - Harwich International 06:45 - 08:11
- Stena Line ferry Harwich - Hoek van Holland 09:00 - 17:15 61.40€ incl. Eurail discount
- subway to Schiedam
- Schiedam - Amsterdam frequent train
Book ASAP option 3 to avoid getting stuck in the UK.
Munich - Rome: few routes possible. At least one mandatory reservation, supplement on the Brenner route.
https://www.seat61.com/interrail-and-eurail-reservations.htm