Im not an expert in the period, but could the Goths have migrated to Great Britain?
Technically: there's nothing impossible, altough that rather than migrating, you could see a Gothic laeti (sort of deportation of groups of people Roman practiced) in Roman Britain for some reason with a strong Roman Empire.
And leati tended to be absorbated by whoever settled in numbers (as it happened with Alans laeti or small foedi IOTL in Great Britain), so either you have Anglo-Saxons taking the lead (remember that the Saxon Shore is already a thing by the IIIrd century), either it knows the fate of provincial laeti (such as in Gaul) and while influencing culturally the neihgbouring region (which provided to be a great assest for cultural integration, then assimilation of Barbarians in the IVth and Vth centuries) doesn't create a Gothic identity.
That said, it's extremely unlikely. Goths as a people first appeared along the Roman limes in Danube and the shores of western Black Sea in the IIIrd century. Their ethnogenesis, how they appeared as a people from a melting pot of Vistulan Germans, Basternae, Dacians, Sarmatians, Romans under the pervasive influence of the Late Empire, made them right from the start leaning to this region.
Migrating from Danube to Britain would have been a bit weird and, frankly, not making a lot of sense : either a IVth century migration as Huns arrived that would somehow neglectg the obvious way out in eastern Romania would involve migrating whole chiefdoms inside Barbaricum (Sarmatians, Marcomanni, Suevi, Franks, etc.) until reaching the North Sea and pulling a sea transport rivaling with already established Saxoni, Eutii, Angli, etc. would at the very best be hard, especially when Roman policy in Barbaricum was to support whoever promised a bit of stability with money, weapons and good will.
A migration to Britain in the Vth century makes even less sense, as Roman Brittania was clearly a backwater territory that even Romans tought being dispensable as garrisons were gradually removed until the very early Vth : it was not that wealthy, regularily raided over, with several estabishments (Scoti, Saxons, Votodoni, etc.) already there. Compared to settle in litterally any other Roman province, it was the worst choice : while you could argue that Northern Gaul was as well harboring several settlements (Saxons, Eutii, Alans, Alemani, Franci, Brittons, etc.) the territory was far more structured (which allowed an easier regional hegemony,as did Franks as having the legal imperium over Belgica) and more wealthy.
In all fairness, I could arguably see (with significant changes in Late Roman history) a northern Gaul based Gothic branch (in a TL where the Vth century notion of Visigoti or Ostrogoti doesn't exist as such) taking the lead in Northern Gaul, and managing to get a grab at southern Britain. Franks did so partially IOTL (greatly helped by a Frankish presence in southern Britain, and a Jutish presence in Gaul), probably clientelizing and dominating the Justish Kingdom in the VIth. But it was mostly contextual, and even Franks way more favoured a mediterranean and central European policy would it be only to approach better trade roads and the core regions of the Early Medieval era.
To have these Gaulish Goths focusing on Britain instead would ask for a strong power preventing them to do so (I'm thinking of a still existing WRE, for exemple), without any guarantee of result in Britain. But, and that's an important but, you might end with a Gothic identity in southern Britain where some Goths (thanks to the significantly lesser roman influence and presence in the island) could use their language later than they did in Spain IOTL, or even ITTL Gaul.
At this point, I must stress that migration of peoples in the Late Antiquity shouldn't be misunderstood.
Maps like these are extremely misleading, if not outright wrong. You didn't have entiere people, as in coherent human groups organized as chiefdom(s) moving from Scandinavia to Spain as the same culturally, politically, institutionally or else.
While you certainly had large groups moving from Vistula to Danube (for what matters Goths) and possibly calling themselves Guts, we must not think of them as Goths in the sense we give to Trevingi, Greutingi, Visigoti, etc. (A bit like New Yorkers aren't the result of a mass migration from Yorkshire, if you will)
While a migration from these Vistulans Germans to the west would be weird (for reasons aformentioned), it could technically happen, and we could see them, rather than being absorbated by western Germanic confederations, forging one.
That said, they would be, for all purpose and intent, a different Gothic people : western Germanic, culturally and structurally closer to Rehinish confederations as Burgundians (which IOTL might be the result of eastern Germans migrating into the west, and foring a new confederation, a western-based one), Franks, Suevi, Saxons, etc.
But would they have been able to do what the Anglo-Saxons did and not only keep their language, but eventually assimilate most Celtic and Latin-speaking people? That's what I'm not so certain about.
Given the backwardness of Britain, the lesser Roman influence and structuration of the island, and the over-reliance on Barbarian foedi, in this hypothetical scenario, I wouldn't see why not.
It doesn't mean Britain wasn't more or less romanized, but we have to remember that romanization is much more of a creolisation, that could take different depths : in Gaul or Spain were pre-existed political and cultural super-structures that Romans could take on, it went the way of a quick and deep romanisation. In Britain, not so much (giving that Romans barely controlled the westerns parts of the province to begin with).
Regardless of the likelyness of Goths settling in Britain (see above), I'd guess they'd go the same way of Angli, Saxoni, Jutii, etc. as creating a whole array of tribal kingdoms only slowly forming cyclical chiefdoms : a cyclical chiefdom, roughly, is what happen when a primitive state manage to takes the lead onto the ensemble of its neighbours and roughly unify them; then crumbles for the lack of structure to support it, repeat ad nauseam until the result of unifications are strong enough to solidify the unity. Now, I really think that ITTL, we'd have western Germanic Goths, making the cultural difference with IOTL Barbarian peoples in Britain meager at best.
Or Andalusia (Vandalicia), which was only actually ruled by the Vandals for about twenty years.
It may be more of a case of medieval folk ethymology there as Vandalicia or Vandalusia being transitional forms we never see appears in contemporary texts between the Vth and the VIIIth (at the contrary of a lot of others, such as Gothia Lunga for Catalonia) : apparently it doesn't make a lot of phonologic sense as well with Andalus.
I'd rather be favouring the hypothesis about
(Guthiudda) landahlauts, an hypothetical Gothic name (Gothic lot, implied the land) that would have gaven birth to the (attested)
Gothica sors, which while not making reference to southern Sain (it would rather be Betica), was used. If the hypothetical Gothic expression survived relativelt late (while the Gothic language in Spain was most probably extinct, ceremonial use of Gothic words and expressions did existed there and there), it could have been borrowed by Arabo-Berbers.