There was a real life emigre Anglo-Saxon military regiment from England, stated as being post-1066 exiles, who ended up fighting in the Byzantine army for the new emperor Alexius I Comnenus at the battle of Dyracchium/ Durazzo in Epirus (modern S Albania) against invading Normans from Naples/ Sicily (then ruled by count Robert 'Guiscard', ie 'The Cunning') in 1081. Alexius had taken over power in Constantinople by a coup from the junta of aristocratic generals, led by the aged and son-less Nicephorus Botaneites (Emp Nicephorus III) who had deposed the legitimate but incompetent young Emperor Michael Ducas in 1078 as the Seljuk Turks overran Asia Minor, and a pretender claiming to be Michael had turned up in S Italy to ask Robert for help to get his empire back - or so opportunistic Robert claimed. R then invaded Epirus to try to take it and Greece from the preoccupied Byzantines, and possibly to seize the entire Empire.
Alexius left the advancing Turks alone with a truce to take his main army off to Epirus but was defeated by Robert at Durazzo and the English regiment, fighting on foot with axes like Harold II's 'Housecarls' at Hastings in 1066, was defeated and mostly killed by the Norman cavalry. But thanks in part to their help, Robert had too heavy losses to advance quickly as Alexius fled E over the Pindus mountains to Thessalonica, and A was able to hold out with reinforcements when Robert eventually did attack Macedonia and Thessaly; the invasion stalled and Robert had to go home to help Pope Gregory resist a German invasion of Rome. When Robert returned to Greece in 1085, aged c. 70, he then died on Corfu and the invasion ended . (Details in the 'Alexiad', biography of Alexius and how he saved the Empire from Turks, Crusaders, and Normans, by his daughter Anna Comnena, pioneering Byz woman historian: I studied this for my History degree which is how I remember such a lot of it.)
It is not clear if the English were a coherent band of soldiers (plus families) from Harold's army in 1066 who fled London after William the Conqueror took it, and made their way to Constantinople - where many English and Danes and Norwegians (and to confuse matters 'English' from E England who were descended from C9th Viking invaders and who still spoke Danish and had a Danish culture in the C11th) were already in the 'Varangian Guard' regiment. One of the latter's top officers in the 1030s had been the man who Harold II defeated at Stamford Bridge (Yorks) before Hastings as he invaded England, king Harald 'Hardrada' of Norway. The 'Varangians' were originally Russian mercenaries in Byz, loaned to the Empire by their new ally grand duke Vladimir of Kiev (of Viking descent like many of his elite) after their peace-treaty in 988, but a lot of them by the 1080s were English or Danes. The English exiles may have come to Byz directly via the Meditt, or via Russia along the trade-routes down the Dnieper -or have left England after 1066, after the failure of the 1069-70 revolts against King William (which had been helped by king Swein Estrithson of Denmark, Harold II's cousin). If they had survived the battle of Durazzo or even won it, and/ or had more soldiers available from a larger exodus from England, then I can see Alexius sending them off to the Crimea as a garrison to keep it loyal while most of his men were fighting the Normans and Turks in the 1080s.
The southern Crimea (ie the coast S of the mountains, especially the coastal towns led by the capital, 'Cherson' near modern Sevastopol) had been a Byz province since the C5th AD, inherited from the Roman Empire - which had taken it over from the indepedennt Greek settler state of the 'Bosporan Kingdom', which was a series of very ancient Greek colonial trading-cities dating back to the C6th to C3rd BC. (Urbanization in this part of Ukraine/ Russia is thus older than in the UK.) The inland part of Crimea was a tribal area, usually inhabited by the same nomads as the Ukraine steppes until the Goths , moving Westward towards the Roman Empire in the C4th AD, settled it; they then took refuge here in the late C4th and C5th from the nomad Asiatic Huns (who conquered the Gothic kingdom on the steppes after a victory in 376). The Gothic kingdom then merged with the Byz province, and a 'Gothic' ethnic and cultural/ linguistic presence seems to have lasted into the Middle Ages and been noticed by passing European merchants. The Byz-Gothic province of 'Cherson' lasted as a Byz province until probably shortly before the Crusader conquest of the Byz capital in 1204, except for a brief Russian takeover in 988 led by grand duke Vladimir - the first Russian conquest of the Crimea, cf the claim of another Russian ruler called Vladimir in 2014! The Crimea then transferred to the Byz successor state of the empire of Trebizond (aka modern Trabzon, on the SE side of the Black Sea) which ruled the Black Sea area, and later to the incoming Genoese merchants and navy in the C14th; and was taken over by the inland Khanate of the Crimea (part of the nomad Mongol successor states) in 1475 as they evicted the Genoese.
I can see a stronger and coherent 'English' military exiles' colony, set up by Alexius I to bolster the Byz army there in the 1080s, lasting merged with the Byz/ Gothic presence as far as 1475, from 1204 as an ally or vassal of Trebizond - which unlike the central Byz empire pre-1204 would lack the large army needed to conquer any local rebellion. But the stronger Crimean Khans, backed by steppe Mongols, would then probably have the means to subdue it, and to blockade any coastal towns with better artillery into submission long-term - unless there is no Ottoman state ruling all of the Greco-Anatolian region to assist the Khans (their OTl vassals) with their fleet and instead there is a surviving Byz state in Constantinople or a navally powerful Christian Trebizond state . Once the Russians take over the Ukraine steppes (Peter the Great tried this in 1711 and Catherine the Great succeeded in the 1770s) Cherson is easy prey for Russia - but the Anglo-Greek-Gothic community can survive as Russian subjects, used as propaganda for a Greek Orthodox 'Crusade' mission by Catherine as she attacks the Turks.