From Kaori O'Connor's Cuisine, Nationality and the Making of a National Meal: The English Breakfast: "Large cooked breakfasts do not figure in English life and letters until the 19th century when they appeared with dramatic suddenness" (p160). The industrial revolution triggered a number of social changes, including when people did things - Breakfast became way of starting the day instead of something one ate a leisurely pace at some point mid-morning as the nation moved from a two-meal system to a three-meal system.

"When a new edition of the work now called Mrs Beeton’s Book of Household Management appeared in 1888, it had a new extensive chapter on breakfast including menus, recipes and table-settings for an English country house breakfast" (p166). This is a stark contrast from the 1865 edition which recommended cold cuts and, basically, leftovers. The Victorian English breakfast was a fairly flexible thing and not really what we'd call a full English today and this continued for some time:

I always remember what a great feature was made of the breakfasts at my grandfather’s house parties at Port Eliot, and of the numerous courses that succeeded each other. There would be a choice of fish, fried eggs and crisp bacon, a variety of egg dishes, omelettes and sizzling sausages and bacon. During the shooting parties, hot game and grilled pheasants always appeared on the breakfast menu but were served of course with-out any vegetables. On a side table was always to be found a choice of cold viands; delicious home-smoked hams, pressed meats, one of the large raised pies for which Mrs Vaughan (the cook) was justly famous, consisting of cold game and galantine, with aspic jelly. The guests drank either tea or coffee, and there were the invariable accompaniments of home-made rolls, piping hot, and stillroom preserves of apple and quince jelly, and always piled bowls of rich Cornish cream. The meal usually finished with a fruit course of grapes or hothouse peaches and nectarines

From Lady Raglan in White, F. (1932) Good Things in England

This would be at the rather extreme end of things, but it demonstrates the idea of English breakfasts being bountiful and very, very meaty.

Rationing in and after WW2 caused a shift as many foods were scarce or controlled, but in the period after there was a rise in bacon-eggs-chips-and-beans "working class cafés", out of which the modern all day breakfast idea came about. This effectively reshaped and re-codified what people thought of as a full English breakfast.

The term "full English breakfast" is a fairly recent one but that's more linguistic note than a culinary one.

Regarding tomatoes - the 1861 edition of Mrs Beetons' Book of Household Management contains 98 instances of the word tomato, suggesting they were fairly well integrated into English cooking by that point.

tldr: The "Full English Breakfast" as a semi-defined dish/meal is basically a Victorian innovation and wasn't really "standardised" until after WW2.