London to Bayeux

The only method that's all rail is the reserved Eurostar train from London to Paris Gare du Nord and then catch an unreserved train from Paris Gare St. Lazare to Bayeux. Transfer between train stations by taxi, bus, or metro. Book the Eurostar months ahead for the best ticket prices.

Looking at Rome2Rio for details about the options; none of them look fast or easy, so I don't see a clear "winner." Check prices on the actual websites of the operators (train company, airline etc), as prices aren't always correct on Rome2Rio.

https://www.rome2rio.com/s/London/Bayeux-France

The Bayeux Tapestry is amazing. Before you see the Tapestry, read the book about the Tapestry that explains each portion of the tapestry and the history behind it.
https://www.amazon.com/1066-Hidden-History-Bayeux-Tapestry/dp/0802714501/ref=sr_1_1?s=books&ie=UTF8&qid=1486251393&sr=1-1&keywords=1066+bayeux+tapestry

There are also narratives in English all along the tapestry to explain what you are looking at, but I agree reading a book of the history would be an added plus.
Bayeux is a lovely town and we stayed there 3 days.
(took time to tour the beaches of Normandy and Mount St. Michel)