How to take the thrilling high speed rail journey between these cities

Take this route via Paris and you will need to make the transfer between the Gare De Lyon, where the Lyria train will arrive, and the Gare Du Nord, from where the Eurostar will depart.

On the usual timetable the first six departures of the day to Paris all connect into Eurostar trains on to London, which depart from the Gare Du Nord around 1hr 30 mins after the Lyria train will have arrived into the Gare de Lyon.

Checking-in at the Gare Du Nord:

At the Gare Du Nord the check in and waiting area for the Eurostar (the 'Hall Du Londres') is at an upper level, above the main concourse.
The access to this upper level is some distance from the voie/platforms/tracks that the Eurostars depart from.
The escalators leading up to Eurostar departures are located at the rear wall of the station building, on the opposite side of the concourse to voies/platforms/tracks 14 - 15.

When the service is operating normally, the transfer across Paris by the RER train will take less than 25 mins, from stepping off the TGV at the Gare De Lyon to being on the concourse at the Gare Du Nord.
So if the train from Basel arrives in Paris on time and the RER is as it should be, you will be at the Gare Du Nord around one hour prior to the departure of the Eurostar.

However, Eurostar recommends that holders of Standard Class And Standard Premier tickets should arrive at the Gare Du Nord 90 minutes before departure.
Which may explain why Trainline doesn't typically sell these fastest journey options.
Also if the train is from Geneve is delayed by 30mins, making the connection into the Eurostar will become a race against the clock.

Though as access to Eurostar is closed off from 30 mins prior to departure, being at the Gare Du Nord one hour before departure will allow enough time to make the Eurostar.

Eurostar recommends that holders of Business Premier tickets should arrive at the Gare Du Nord 30 minutes before departure, with access to check-in being closed off from 15 mins prior to departure.

The Eurostar on to London:

Final Destination: London St Pancras

Note that these trains will no longer be calling at Ashford International or Ebbsfleet International, until an unspecified date likely to be in 2025 at the earliest.

Note that London is in a different time zone to mainland Europe, it is one hour behind; so the journey time is an hour shorter than the time shown by the ticket agents.

Our guide to arriving by Eurostar in London is here.

Book Early And Save: Yes

Online bookings open: up to 6 months ahead of of the travel date

Eurostar does not sell tickets for this journey.

Trainline tends not sell these fastest end-to-end journeys, though sells options which only involve taking this combination of Lyria + TGV trains - but which involve taking a later train on from Paris, thereby de-risking the possibility of having to re-book the Eurostar (see below).
Look for the journeys on Trainline which have the TGV Lyria and Eurostar logos.
On Trainline the main station in Geneve/Geneva is named 'Genéve-Cornavin'.

Seat reservations are automatically included when booking tickets for both trains.

Your ticket will not include the RER train across Paris from the Gare De Lyon to the Gare Du Nord.

Making the connection in Paris:

There is relatively new E.U. legislation which protects travellers against financial loss in the event of missed connections due to train delays, but it only applies to 'through tickets'.
The Eurostar terms and conditions specifically state that when booking journeys which combine Eurostar + Thalys trains 'through tickets' will be issued.
However, 'for all other journeys that combine a Eurostar Service and a service provided by another carrier(s), including when purchased in a single commercial transaction, those tickets will be separate contracts and issues regarding delays, missed connections, cancellations, compensation, and the management of aftersales will be handled accordingly'.

Though Lyria and Eurostar are members of The Railteam Alliance which offers a Hop on the next available train (HOTNAT) policy.
It 'allows travelers to take the next high-speed service leaving from the same station as originally planned when a delay on or cancellation of a preceding Railteam member’s high-speed service prevents them from making their originally-planned connection. This service is free of charge and is subject to the following conditions:

• HOTNAT only applies at the station at which the passenger originally planned to change trains and in cases of train delays and train cancellations.
• The approval of HOTNAT is subject to the available capacity on board each train. Seats are not guaranteed'.

So 'issues regarding missed connection and cancellations' should be 'handled' by Railteam's HOTNAT policy.

Though HOTNAT can't be applied if the arrival into Paris is after the final Eurostar of the day has departed for London.

Much more info is available on this ticketing guide.