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Londinium, Britannia, 284 AD. The military commander Carausius is leading a movement to take Britannia out of the Imperium Romanum. He thinks there is a conspiracy between locals and foreigners to take control of power in Roma, orchestrated mainly by the new sect self-denominated as Christiani, suspiciously enough, founded by and composed of Jews, but also of many Greeks, both groups aiming to recover their glories of the past and remove Roman domination.

Carausius calls for a referendum, under the slogans recupera imperium (take back control) and age Britanniam magnam iterum (make Britain great again). Vote Leave (Sententia Abire) wins.

What can we call this result in Latin? One method is to adopt current practices to Latin. The other is to come up with something idiosyncratic. I cannot think of anything for the latter, but for the former, one option could be Brexire, which is the composition of both Britannia and the verb "to exit", exire. But is this the most appropriate meaning? Alternatives?

The word "Brexit" is a noun, meaning "the exit of Britain from the EU".
The noun "exit" is exitus, fourth declension.
Therefore a natural analogue of the English "Brexit" would be Brexitus.
There might be some use for a verb brexire, but I believe the noun is most relevant.
I have seen "Brexit" used a number of times, and in all cases it has been a noun — read and write complete sentences about Brexit to see how it behaves.¹

I think it is very important that the word is recognized well, so I urge you to use some form of Br[itannia] and exire/exitus.
If you use a different word, you are better off explaining the whole thing.
However, my impression is that such portmanteaus would not be idiomatic third century Latin, so you would need to explain (i.e., say "the British exit" or something) instead of coining a new word (like "Brexit").
I think words like Brabitus or Bregressio are way too hard to understand, and are therefore not that good choices for ordinary communication.
I can't keep you from saying Brexodus, but I will struggle to understand if you do so (outside this question where I know to expect things like that).

Nuntii Latini had a news item on Brexit, but it makes no use of the word (even though they rely on exitus):

Suffragium de exitu Britanniae procrastinatum

Theresa May, prima ministra Britanniae decrevit, ut suffragium, quod die Martis (11.12.) de exitu Britanniae in parlamento Britanniae fieri debebat, procrastinaretur. Tum pars delegatorum factionis conservativae flagitavit, ut de fiducia illius factio suffragaretur, sed suffragio facto May votis ducentis contra centum septendecim (200-117) superior discessit.

Also, this use in Nuntii Latini confirms that exitus is a decent choice for this kind of an exit.

¹
Consider for example: "When will Brexit happen?" "What will May do if Brexit fails?"
The word here is a noun, and I have yet to see it used as a verb in the news.
Phrases like nolite brexire are possible but come across humorous and weird.
I wouldn't say "don't brexit" in English.

I pass on (without comment on the politics involved) the following letter from Dr. David Butterfield of Queen's College, Cambridge to the Editor of the Daily Telegraph:

"Quid sibi vult Brexit? qui Brexit ipse est?
quomodo, qua, quando terra Britannica Bregat? nunc post lustra novem Bregere — an Brexire? — necesse est: parsne fuit Remanes parte abeunte minor? vox populi, divi est; at vox repetita fit echo. num referenda iterum sunt referenda plebi? est gravius multo spectare futura Britannis: quid refert nobis extera vita novi? quaerere plura queam; meliore intendere malo: Europhilos ut nos dedecet usque queri.
nam, Brecta EUropa, non parvam noster amoris stillam Euro paean inde favente teget."

Given the ambiguity of "exitus" (departure, death) I should think that "Brexitus" is absolutely perfect.

Considering Shakespeare and all that, the answer is Brexeunt, surely ;-)

canis est scriptor prandium

Literally, "dog's breakfast".