alternatehistory.com

[this is part of a much bigger alternate history project, so please concentrate on the question, not the validity or probablility of the context]

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This project started from about the year 1070, and it is only realistic until about 1200. The next story is mostly a work of fiction and imagination (and you have't seen anything yet. Not only that this story is overly-simplified, it is also unfinished, as I plan to keep imagining the continuation based on the answers I receive here) . I only need the answer because I don't want to stray too much from realism (more than I already did).
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It is the year 1300. A british-ish empire spans over all of western Europe except a small independent Papal State (basically, if it's west from Warsaw, it's in the Empire), North Africa, Arabia and even colonies in India. the whole territory is divided in duchies and counties, pretty much like a feudal kingdom, but much bigger and with a nobility class that also acts as bureaucrats. The Balkans and Northern Middle-East are part of a much weaker, but incedibly imperialistic Byzantine Empire (who call themselves the New Roman Empire, no longer considering themselves a continuation of the SPQR, but a separate, greek reimagination of it), rival to the British Empire.

The British Empire is quite culturally united. The British Isles (even including Scotland!), France and Spain are authentically english, while the rest of the empire is very diverse, but peaceful. There are even ruling dukes inside the empire who are ethnically arab or indian (but culturally european). The whole empire is catholic, but because the whole catholic world is under the Empire's rule, the Papacy is weak and its powers are purely symbolic, the Empire giving priority to economy, military, wellness and only then religion.
Yet, even with the Empire's tolerant and inclusive attitude, everything outside of the empire is seen as space in need of civilisation and colonisation. Due to Imperial and Byzantine conquests, The whole middle-east is christian, and islam is (except for parts of West and East Africa) gone.

The technology of the empire is 150 year ahead (1450's level by 1300), and the world, in this timeline, due to recovery of Ancient Greek knowledge by the Byzantines, is known to be spherical, and having more continents (the approximate measurements of the Globe and of Eurasia and Africa are known, and the Greeks and British simply do not believe that the rest can be an empty ocean).

Now, given these informations (sorry for the wall of text), when would the Europeans discover the Americas (and which part of it exactly) and what would happen afterwards?

Also, what chances would the Byzantines have to get their own colonial empire (not necessarily in America) afterwards, given that they are blocked on all sides by the Empire?

The map of the World is attached.
Blue is The Empire
Red, the Byzantines
Yellow, least important, is a Zoroastrian Persia, allied to the Empire, because the Empire wants someone else to take care of the Byzantines (long story, would also be interested of seeing its chances of having an empire)
The rest of the world is just the way it was in 1300.

map.png
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