Why nightrains probably will never run through the Channel Tunnel to the UK :
- very special (safety) requirements for trains passing the tunnel
- UK border control facilities at every boarding stop necessary
- high track access fees https://twitter.com/WiredUK/status/1294212516864819200

30 years ago the national railways BR, SNCF, SMCB and NS had actually plans for night trains and rolling stock aready existed.
This interesting article, written by a former NS employee, describes the history of the project and explains why it failed:
http://swiss.nwrail.org.uk/nightstar.pdf 

And 40 years ago there was a direct London/Brussels-Paris night train, the "Night Ferry".
It's last run was on october 31st 1980.

https://en.wikipedia.org/wiki/Night_Ferry

More interesting material about the "Night Ferry": http://www.irps-wl.org.uk/wagon-lits/services/nightferry
Arriving Paris Gare du Nord on the last "Nioght Ferry" on nov. 1st 1980:

So, the "Night Ferry" is gone. Night trains through the tunnel remain a dream, using regular EST-trains as connecting service is also problematic (ticketing/pricing).
What can be done?
Maybe establishing dedicated coordinated trains with a Lille-transfer-hub?
@seatsixtyone

A E320-trainset has ~900 seats.
Such a train could be a feeder into 4 7-car-nighttrain-portion: 2 trains with 2 portions with each 250-300 places, some places remain for "intra-continental"-passengers.
For example Lille - Hamburg/Berlin and Lille - Zürich/Munich?

The transfer situation in Lille is not ideal, as the stations Lille-Europe and Lille-Flandres are some 250m apart, but I think it would nevertheless make sense to establish such a transer hub as close to London as possible.

Establishing the transfer at Calais-Frethun could be an alternative, but probably it would be necessary to built additional tracks to park the trainsets during the day/night....