Like on Thalys and some cross-border TGV there is a limited number of seats designated for Interrail/ Eurail pass holders on Eurostar. If that bucket is full, you cannot buy a reservation for Interrail/Eurail any more. You can book an ordinary ticket or check other trains.

When are you travelling?

If everything is booked, you could check if there’s still pass holder seats available to Lille. From Lille you have plenty of TGVs to Paris, that don’t have a limited number of pass holder. A reservation will cost 10 or 20 EUR.

You can book domestic trains in France via the travel.b-europe website (4€ booking fee per booking, Pass Cover Number needed, request it here) 

1. The number of pass holder seats on Eurostar is limited. They sell out before the train is full.
2. Eurostar removed that possibility from their website. The best website to check is:
   https://www.b-europe.com/EN/Booking/Pass#TravelWish
   It will only show trains with available pass holder seats. Click on Earlier/Later trains until you find trains. If you don't find seats to Paris, then try Lille.
   You need a pass cover number to book. If you have a mobile pass, then you can generate one:

    

The date we go is December 28.

In the middle of the Holidays there’s serious demand.

I don’t find any connections on that date. The first possible to Lille is at 31st of December, direct to Paris already 1st of January.

You might want to look into ferries (Harwich-Rotterdam or Dover-Calais) and then continue to Paris by train. Or buy ordinary Eurostar tickets. Or Fly (although it’s like swearing out loud in a church to suggest this on a train forum )

In 1st class, there's availability to Lille and Brussels leaving London at 8:55.

OR-if on budget and not prejudiced too much about that: BUS - FLIX or the quaintly named BlaBla. Goes via ferry, occasionally even via tunneltrain. Also overnite-but you wont get sleep. With some luck even cheaper for ticket as just the surcharge for €*

For the train-ferry-train combo read seat61.com -as only a very few such ferries still allow that, daytime only. Ferryticket will then certainly cost more as otherwise €* supplmt

Don’t try the Dover Calais ferry option. I’ve done it and it is a time consuming nightmare. Check out Flixbus London to Paris if you can’t get a cheap flight (very unlikely you’ll get a flight though)

Ended up booking a flight to Zurich which is  where we were headed.  Sorry people Didn’t realise travelling that time of year would be so hard. This is good information though as my daughter is trying to get from London to Paris on boxing day.