Sorry, but unless you want to spend more time in transit than actually seeing places, you need to add additional days or reduce your itinerary. Two nights somewhere means one day there, particularly as you have so much travelling between your locations.
Holy Island is the geographical outlier and access on and off the island is dependent on the tides, so you may have to switch your dates around to make this possible.
You are right to be concerned about driving immediately upon arrival whilst jet lagged - please don’t! Have the first night in London or at the airport, depending on your arrival time. Alternatively, if you are up to it, take the train straight to Carmarthen - it’s a long four and a half hours + on board, with changes in London and Newport. Trains run hourly, so if you just miss one, it’s a long wait! I am not sure I would want that journey after a long flight.
Carmarthen to Shrewsbury - likely to take the best part of 4 hours driving including finding somewhere to park etc. If you can get one of the direct trains, the door to door time will be about 4.5 hours.
Shrewsbury to Holy Island - a nightmare by train, involving between 2 and 4 trains depending on the time of travel, then a bus from Berwick upon Tweed. There is only one bus a day - the times vary depending on the tide times and therefore you may need to get a taxi. Likely journey time door to door is 8+ hours. Driving is nearly 300 miles and is likely to take you 6.5 hours + assuming you don’t stop or get held up in traffic. These times assume no delay for crossing the tidal causeway. Today, you can’t access the island between 14.55 and 20.00, so if you got there at 15.30, you would have a long wait!
I am used to the U.K. roads and wouldn’t relish doing this much driving in such a short timescale, but I wouldn’t want to spend so long on trains either.