"Cycling the E2E" has covered every sort of machine imaginable from unicycles and upwards! - just on our 2010 LEJOG we saw full-on carbon machines, Dawes and Thorn tourers, Trek hybrids, a couple of "bottom-of-the range BOGOF newspaper ad" machines, a tandem or two, two recumbents and a Brompton folder - they all seemed to be going OK.
Obviously the type of ride, as considered elsewhere in the FAQs, has an impact on the bike - the key issue being that you can ride it with some comfort for the miles you plan to cover, day-after-day.
The most popular machines appear to be "touring" or "hybrid" bikes - one person's definition here may differ from another ... Clearly - if you are riding, fully-supported, in 6 days you'll be going for something a bit more sporty ... the 4 week camping tour via the Hebrides will call for a heavy-duty machine.
It's horses (bikes) for courses.
My own rides were on a Dawes Galaxy - originally built in 1975 but upgraded to current standard components.