There is probably some Anglo-Saxon settlement (nebulous as the term may be) in Britain but maybe it could "disappear" like in Northern Gaul. It could be significant for a few decades but then be absorbed by the larger population. Maybe there is only a "Jutish" kingdom in Kent with the rest being Romano-Briton ruled polities.
Thing is, Gaul is ruled by a relatively strong state, compared to Britto-Romans polities : Frankish hegemony in Gaul didn't made disappear Armorican Saxons (we have specific mentions up to Carolingians), but it did put them under a rug and may have motivated some south/north migration or at the least demotivated further settlement in Gaul.
Even if, for some reason, Scandinavian and North-West Germanic peoples doesn't settle in most of Britain (and we should find a place where Scandinavians at least settle instead), the Channel itself is going to recieve a significant migration, because it was an obvious way.
Well, if the Angles, Saxons, Jutes, et al. don't form sub-kingdoms and instead join into a larger confederation similar to the Obotrites, their identity could solidify in response to German expansion later on.
That"s unlikely : I mentioned Euthiones earlier that were a Jutish group settling along Saxons in the continent : they were quickly integrated within Saxons because all these peoples weren't really quite differenciated either linguistically or materially. Especially when it comes to heterogenous familial groups settling in relatively few numbers (remember that early medieval Britain isn't really a demographic powerhouse, to say the least), they are likely going to be integrated in a Saxon or Thuringian ensemble at best as its periphery.
Arguably, I forgot about Thuringia in my above post, which I shouldn't have giving their known proximity with both Saxons from one hand, and with Angli and Varini in the other hand. It's equally likely that Thuringians would get the upper hand on Anglians/Jutish/Saxonic groups on the other side of the Elbe.
East Franks and Germans had difficulty subduing the area east of the Elbe for a long time.
That's debatable how much they didn't : we simply don't have much sources on it, except the area was poorly inhabited until Wendes kicked in and merged with the remaining population. At the very least, arguing that before Wendish migrations the area was badly controlled is entirely speculative.
Avars would also present a significant pressure and influence along with the Slavic interaction.
Not this much north of the Pannonian plain : wjile Avars mostly formed Slavic chiefdoms at its limits, it never really went besides the Carpathians or Bohemian mountains IOTL.
We could also see more Geatish migration, adding more North Germanic elements to make it more "foreign" for the East Franks.
Foreigness is a bit of a non-problem : Franks mostly managed to integrate into their German sphere of influence a variety of peoples : Bavarians, Thuringians, Alamans, etc. and at times Saxons and Jutes in England. Of course, the appearance of Eastern Franks is a much later event, that might not happen ITTL giving the PoD (altough I agree that it's not particularily butterfliable as such, we're talking of the situation in the 500's-700's and not 850's).
As for a more important Gaetish (probably Danish in this context) migration : again, the obvious way is Britain, not Baltics. But even if it was, well, it would happen the same thing that with Scandinavian elements in VIth century England, they would get merged with North-Western Germans because the differenciation at this point shouldn't be exaggerated. TTL Saxony would be remarkably similar to IOTL Saxony in last analysis.
If it's conquered by the Franks on the same timeline as the original Saxony, then the Holy Roman Empire will be substantially more powerful and the Ostsiedlung much more extensive, with a starting point farther to the east.
Carolingian takeover is far from being obvious, ITTL (without talking about Ottonians and HRE), so I think we'll stick to the VIth to VIIIth centuries : Saxony was half-part of the Merovingian sphere, at least in its periods of greater hegemny (half of the VIth, half of the VIIth) and it did have an impact on characterisation (politically wise) of the region. While I wouldn't see a Merovingian conquest of the whole of Saxony ITTL either (although they did took some part of Saxony IOTL), I'd suspect that Thuringia would still befall to Merovingians ITTL including their periphery on the other side of the Elbe.
According to Wladyslaw Duczko in Viking Rus: Studies on the Presence of Scandinavians in Eastern Europe, there was a higher number of items of Scandinavian material culture in the east than in the west, probably indicating a heavier Norse presence overall.
It could as well be because of a better cultural mixing-up and acculturation : Danelaw England had significants finds of Scandinavian material culture, but we know that Anglo-Scandinavian people was composed in a very large part of Anglo-Saxon population.