After some research: I think the London - Amsterdam is the annoying bit as the quota is sold out on your travel dates. Paris - London has plenty of Eurail availability, same for London - Brussels.
Again from a quick look, tickets seem cheaper:
- Paris - London is around 100€ on 14th February
- London - Amsterdam around 200€ on 16-17th Feb
London - Amsterdam is definitely on the expensive side but it's an extremely popular route (way more convenient than the plane) so trains fill out. Eurostar also has a monopoly until further notice.
Alternatively let me also mention the (hidden) Eurostar Snap offer. It is much cheaper than regular tickets but you need to be a bit flexible, perhaps not possible on such a short trip. See https://snap.eurostar.com/rw-en
With that offer you might need to book London - Brussels and pay Brussels - Amsterdam separately (non-Eurostar trains exist and would be cheaper on that route). Or trains might not be available at all during that Valentine's weekend and esp. during Biritish school holidays... who knows. Their goal is to fill less busy trains.
Don't hesitate to ask any questions, we'll be glad to answer. :)
Maybe mention the precise route and date and we can advise if there are alternatives. Like London - Brussels and changing there or even the UK-NL overnight ferry if you don't have accommodation booked.