I've lived all over Britain, born in the Midlands, worked in the West Country, South Wales, Scotland, and London/Home Counties. I think the Full English can consist optionally of any of the following:
Egg - preferably fried with a soft yolk, but "well-done" fried and arguably scrambled (with plenty of butter) are acceptable. Poached, or other dodgy new-age garbage, is clearly unacceptable.
Bacon - either back or streaky, fried in preference to grilling.
Sausage - preferably good pork sausage, fried in preference to grilling.
Black pudding - definite place for a good piece of fried black pudding. White pudding seems a little too regional to qualify.
Fried mushrooms and tomato - common across much of the UK, and perfectly valid additions.
Bubble and Squeak - popular in parts of Britain. Probably the only other vegetable dish that can be on the plate without losing focus. Hash Browns are definitely too American to qualify, as are things like potato waffles.
Fried bread - for me one of the killer features of a real English Breakfast. Make sure it is fried in the fat that has dripped out of the sausages. It should also be fried on both sides, no one-sided French garbage.
The following though are marginal:
Toast - toast with marmite or jam or marmalade might be yummy enough, but seems distinctly out of place as part of a full english. I think you can argue reasonably for it as a starter or pudding before or after your full english, but not at the same time.
Baked beans - no - seems a little too American for my liking, and strictly probably descends from Cassoulet, making it too French for a "Full English". Also, by not being fried, it doesn't deserve a place in a quality fry-up.
The key criteria for me for a full English is that you should be able to cook it all at the same time in the same frying pan and in the same lard (or dripping at a push).