Just to be clear to everyone, when I say a Celtic identity, a Celtic culture, or a Celtic artefact, I am mainly referring to an island that identifies mostly with its Celtic origins, rather than with the origins that arose as part of later migrations. When I say 'Celtic', I do not intend to refer to all the separate Celtic cultures under a single umbrella; as I said earlier, there could be several nations, several nationalisms, all with common roots in the pre-Roman Celts.
As far as I can see, there are two issues that need to be addressed: reducing the influence of the Romans to ensure a stronger Celtic culture in Britain; and enabling this Celtic culture to be better able to resist the Germanic incursions.
If we manage that, then the culture is likely to be strong enough to withstand the later Scandinavian incursions.
By no stretch of the imagination am I an expert in this period, so treat the following with all due caution.
There are a lot of butterflies that would arise from no Roman invasion of Britain, to the extent that one would have to look at the impact on Gaul, and on Rome itself. To avoid this, and simplify our job, let the Roman invasion go ahead.
The main resistance to the initial Roman invasion in AD43 was by the Catuvellauni, involving what is described by unreliable sources as a two-day battle for the Romans to cross the River Medway. At the time, the Catuvellauni were fighting alone, because this was taking place during a “period of turmoil”, and the other Celtic tribes were only too happy to see the Romans chew up the Catuvellauni.
This is where we can change things. Bring the Celtic turmoil forward a year or two, and by 43AD, the tribes have kissed and made-up, and when the Romans come, other tribes come to the aid of the Catuvellauni, and the Roman invasion bounces after hard fighting.
Claudius tries again in AD44, with much the same result, this time with the Celts deciding to attack as the Romans are getting out of the boats.
By now, the Roman military is getting rather tired of facing British Celts. Luckily for the Romans, the Celtic inability to keep the peace between the tribes comes to the fore, and when the Romans invade in 45AD, they face a disunited enemy, and OTL result of 43AD comes about.
Romans settle down to rule in southern Britain, controlling up to (but not including) Devon, Wales, and Yorkshire. Influence spreads a bit further through trade and so on. The Roman military is a bit more wary of the Celtic warriors, and come to the predictable conclusion: “We want those bastards fighting for us.” Rather than disarming the tribes and stopping the constant raiding by force, they co-opt warriors into the army as auxiliary units. These do a fair amount of going off to far-off foreign fields, and meeting interesting local people, serving out their time and then coming home, often with goods and chattels and riches.
This is going to add a lot of little snippets of other cultural influences, which is likely to vary widely by region, but overall, the Celtic culture is dominant. The Romans are more cautious about offending the local culture.
In due course, the legions go home to Rome. The Celtic auxiliaries there decide to stay put, and no-one fancies persuading them otherwise. A few years later, the Germanic incursions start, but face significantly stronger opposition. They manage to make some inroads, essentially south east England and East Anglia, but that’s about it.
Northern England is much more Celtic, and there is a stronger tradition of fighting people with funny accents, so when the Norse come calling, they face stiffer resistance. There is still Norse influence settling down, but again, it is much more confined to the fringes.
We’ve got a situation where there are huge cultural variations throughout the country, which generally consist of Celtic + x. Celtic and Norse in Northumbria; Celtic and Angle in Anglia; Celtic and Saxon and Jute in south-east England; Celtic and Roman in southern England; and so on.
Thereafter, who knows?
As I say, this isn’t my period, so it may very easily be ASB territory, but I’m sure someone cleverer than me can tidy things up.