Okay, it's pretty much agreed upon that Britannia is among the worst and useless acquisitions Rome has made in its history
Well...There's eastern Dacia.
Seriously, though, it was less interesting in comparison of other provinces : Spain was more or less Ancient Mediterranean Peru, Gaul a demographical behemoth (meaning big market and big production), Africa a living cash registrer...
So, yes, Britain paled a bit of the comparison, but wasn't that of the trash can you describe.
It's a given that an independent Britain would certainly do better against barbarian settlement and a large amount of the OTL settlers were mercenaries who are likely to never make the trip with no one to pay.
Nope, and nope.
Without a Roman Britain, no unified rule to speak of, or variation around high-kingship as in medieval Wales, Ireland or Scotland. Giving that political disunity was hardly a strong point when it came to scandinavian settlements, I'm afraid that Western Germans would have an even easier time coming in, and right from the IIIrd century instead of the Vth.
In fact, the pauperisation that plagued Germanic society after the fall of Rome would certainly damage a lot the Britton kingdoms, hooked on Roman trade since centuries, making them lot really unstable. To think that no one would have the use for mercenaries because they're not Romans is quite naive, to be bluntly honest.
But, would that better defense lead to more barbarians being diverted south into Roman territory ultimately making the burden of foreigners worse for the empire?
There's no way a divided up and even less develloped Britain would do better against Germanic takeover when doing as much would already be a good thing.
The more interesting question would be about the possibility of non-Brythonic celtic raids from
Ireland and Scotland as IOTL..
But as I said above, there's a certainty that Britain would know the same pauperisation and destructurisation than knew Germans IOTL by the IVth and IIIrd centuries, which would probably have same effects (as in more warfare and plunder, with some peoples migrating and/or expanding at the expense of others).
For consequences on Rome, it would certainly look like as what happened in the Vth, meaning Saxons using Britain as a jumpgate to raid Gaul, but giving the really limited numbers involved in the Great Invasions, it would be far from a horde of Barbarians overruning the Romans.
As for the effect of migrations :
- Western Germans would probably settle by the IVth century in Eastern Britain. Either by their own, or "invited" by Brittons to fend off possible northern and/or western invasions.
- Britain may look more Gaelic, Pictish or northern Brythonic in West and East, with survival of Brythonic languages being in question. (For exemple, maybe Gaelic not only expanding on modern Scotland but Wales and further)