My personal recommendation would be to take the TER200 at 13h21 from Bâle SNCF, arriving into Strasbourg: 14h39 and changing onto the TGV to Brussel, departing: 14h59, arriving at Lille Europe: 17h57 . This is a safe connection as these are the official "replacement" services for the twice daily direct Basel-Brussel EuroCity connections; EC 90 Vauban at 13h21 and EC 96 Iris at 16h21 respectively, which were sadly permanently cancelled in 2016..
From Lille you have a direct ES to London at 18h35, arriving into Saint Pancras: 19h10 local.
Indeed, this is probably the fastest connection at only 6h49m, less than seven hours end-to-end journey time with the help on the new LGV Est Européenne:
Bâle SNCF voie 31: 13h21 (TER 96282)
Strasbourg: 14h39
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Strasbourg: 14h59 (TGV 9894)
Lille Europe: 17h57
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Lille Europe: 18h35 (ES 9153)
London St. Pancras International: 19h10
You can sit in the voiture-pilot on the TER200 -(located at the rear of the train, where the guard is located and is wheelchair accessable), its usually a cross-platform change from this train to the TGV at Strasbourg, but there are lifts located on all platforms should the TGV depart from another, you can make an assistance booking with SNCF anyway as others have suggested above. If you want a relaxing, stress-free journey avoiding Paris then this is the way to go.
In the opposite direction (EC 97), the Iris made London-Basel possible in a day with just one easy change at Brussel Zuid, and was a most enjoyable run through Wallonia, Luxembourg and Lorraine until the train was deleted in April 2016, months before this "replacement" service even started..
Progress...
EDIT: -(it slipped my mind that England is one hour behind and with the time difference it is actually six hours, forty-nine minutes and not five... Dèsolé!

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