Well, it's roughly 3,500 miles from London to New York, assuming any such tunnel - which would have to be a suspended tunnel within the water - was built it would very likely start from somewhere near Weston ( for ease of through running to / from London ) and finish more or less at New York itself.
Now, it is obvious that this would be non stop all the way and would be pretty much straight meaning that you would be on high speed cruise for virtually the entire distance, most likely at next generation TGV speeds of 250mph.
This, assuming the high speed running was only within the tunnel and that Weston to Paddington ( or wherever ) was at present line speeds, would yield a journey time of just over 15 hours.
So if the cost and technological challenges of the construction ( including making it proof against collision with submarines and ships dropping anchor on it ) could be overcome then the journey itself is certainly feasible.
But, and I stress this, it's the money that will be the killer factor. It won't happen!
We also have the interesting effects of the 5 hour time difference as well; a westbound service would cover 10 clock hours while an eastbound would cover 20 clock hours. So an 09.00 departure from London would arrive at New York at 19.00 local the same day but an 09.00 from New York would reach London at 05.00 local the next morning.
This gives a clue as to potential timetabling; departures from London would probably be half hourly between 07.00 and 12.00 for conventional services ( which would arrive between 17.00 and 22.00 local ) and hourly between 20.00 and 23.00 for a Sleeper service ( arriving between 06.00 and 09.00 local ), both types having the same journey duration. That's 15 services.
Eastbound would probably be hourly from between 11.00 and 01.00 ( arriving between 07.00 and 21.00 local ) - but in this direction each service would need a Sleeping element! That's 15 services, so the daily diagrams are balanced but the fleet ultilisation would be relatively low as the timing imbalance requires more trains to be available at each end than you might at first think; they can't just work on a turn and turn about basis.
That throws up a lot of challenges, not least that each rake would probably need to be of couchette type to accomodate the different demands of crossing the time zones in each direction and making sure that both departures and arrivals are at reasonable hours at each end! You'd probably also need a fairly long turnaround for those trains that can make a return trip; possibly as long as two hours.
Let's see what the diagramming fraternity on here can do with those timings...
What would the fleet size need to be?
*throws down gauntlet*