There is a limited number of pass holder seats on the Eurostar and on popular departures and during high season those sell out weeks, and sometimes months in advance.

The best place to see the availability of passholder seats and make reservations on the Eurostar is 

https://www.b-europe.com/EN/Booking/Pass#TravelWish

Press "later trains" and eventually you will find the next available connection. 

If you have a mobile pass you need to generate a Pass Cover Number in order to make the reservation at b-europe. You do that here in the PCN generator:

https://community.eurail.com/news-and-announcements-39/pass-cover-number-generator-is-live-5653

You can check availability from London to Lille and then take a separate train to Paris from Lille. Availability to Lille is usually better than to Paris.

There is basically no passholder availability from mid December onwards. not to Paris, Lille or Brussles, suppressed services from covid, busy time of year and the big sale they ran a few weeks ago has seen most availability drop.

Options are:

Travel another time

Travel somewhere else

Pay for Eurostar separately from their website

Fly

Bus

Train+Ferry, various routes.

1 or 2 you can figure out yourself,

Skyscanner.com is best for European flight fares

Bus or Ferry, ask for details.

For London - Paris there's no availability whatsoever for 4 persons.

For London - Lille, the first availability is on 24 December 6:57 and 12:55.

For London-Brussels, it looks like this:

If you want to book any of those journeys, then do that NOW. Otherwise, look for alternatives.

Thanks for all the help. I've resigned to my fate that to get out of UK its just extra tix directly from Eurostar. I am unable to get seat reservation to anywhere else and the pricey Eurostar tickets seemed to be the most viable option compared to the more pricey flight.. fully booked ferry... the chaotic and long bus ride

Moving along my trip (and learnt my lesson), I have booked my seat reservation from Paris to Geneva using the Eurail portal. Made the payment (pricey IMO), got the email, was informed to printout (which i will). The sense of insecurity just made me checked directly SNCF SNCB but it seems like its not reflected with the train company itself.. is it a cause of worry... is printing out really mandatory.. what if the print out is somehow lost during the trip

You don't necessarily need to print. It's handy to show it if someone is in your seat. A pdf is just fine. They'll be able to scan it anyway. 

TGV Lyria is expensive indeed. You can kinda work around this by taking an ordinary TGV to a French border town and use local trains to cross the border. Can be a bit of a puzzle and takes more time, but less extra costs.

Anyhow, once you're out of France, it means (almost) rid of terribly expensive and mandatory reservations ;) 

You’ll need a barcode on a phone or printout to get through the gates at Gare de Lyon.

@Daniel Li If you share your travel plans in the Community before making any reservations you will get advice on how to travel in the cheapest way. Like not making the reservation from Paris to Geneva, but to the French station before Geneva and then take a local, reservation free, train to Geneva. Then you would only have paid 10 or 20 EUR instead of the very high reservation fee to Geneva.