Not really.

Eurostar cost 30€ and has a limited quota which can sell out weeks in advance. It is indeed a bit expensive but it is the only way to reach the UK by train.

Alternative : ferry between Calais and Dover but it also costs the same and would take a full day

I can find Lille - London for 20€ on Flixbus but for 10€ extra, I’d take the Eurostar all the way (only 2h15)

Afterwards travelling in the UK with a pass is fantastic : lots of trains, no reservations needed (free if you’d like one on long journeys) ! :)

If you plan to go the UK in the summer I recommend tht you book the Eurostar ASAP.

There is a limited number of pass holder seats on the Eurostar between London and mainland Europe. 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

If you don't get any result, 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