If you increase Saxon presence on that side of the channel, I don't think that its a given that the Franks unite Gaul as they did in OTL.
Frankish takeover of northern Gaul isn't really a matter of demographics, but that they were importantly integrated in regional management since the IVth century and critically since the mid Vth century with the rough alliance between Britto-Romans, Gallo-Romans and Franks in the 460's.
Chilperic's interventions were probably considered legitimate enough in Northern-Western Gaul at this point, and Frankish dominance over the region happened too quickly on a relatively decentralized region to be attributed only to sheer conquest.
Saxons on the other hand, were rather minor on this regard and only gained mention with Odoacer (which might have been a distinct Saxon chief,
but also the Odoacer) relatively late : like Alamans, they were too much disunited, too localized and without real connection with the late and post-imperial world to really hope gaining local hegemony.
Saxon migration to the British Isles is roughly in line or slightly earlier than Frankish migration to Gaul from what I can tell
We're talking of really different settlement there : Frankish dominance isn't much a matter of migrations but rather of border peoples that remained in place and when mercenaries got settled within Romania very quickly got integrated into late imperial frames and romanizing extremely quickly. What we call Saxon migration in British Isles is rather actually a whole migration of familial group of North Sea peoples (Norses, Danes, Angles, Saxons, Frisii, Franks, etc.) that Saxons come to dominate and mix with local Britannic population (taking the advantage)
If anything, Frankish presence in Gaul predates Saxon presence in British Isles and Channel by a century and a half, and was more organised and tied to roman politics than what happened in Britain.
It took a long time for all these groups to really be incorporated into "France" and until the 1800s for the cultures to really be incorporated in the "main" French culture.
You're making an hugely anachronistic and ethnicist comparison, to be honest : Bourguignon identity can't really be traced to Burgundians, as much as French identity isn't a Frankish identity : all of these got build much later in the Middle-Ages : if anything, Barbarians groups in Gaul got essentially integrated and undistinguishable materially by the Vth century already, and politically/socially by the VIIth century at latest.
If we take the Normans, while definitely settled by the late Xth, they were integrated in French nobiliar network relatively quickly and before 911 (in fact, it's what allowed them to be integrated politically) and by the XIth they considered themselves as
Franci as can be plainly seen in Bayeux Tapestry
I think it's hard to have the Saxon establish themselves, with a post Frankish foedus in Belgium, over all of Northern Gaul.
True but IOTL, you still had the conception of Saxons in Gaul up to the VIIth century (where they were virtually indistinguishable from the rest of the population), so a larger geo-political space in the lines of IOTL Normandy or Anjou isn't unthinkable.
If we managed to get a stronger Latin presence in Britannia, (maybe Constantine III doesn't strip the garrison); then Normandy may be a more attractive prospect then England.
Probably not : the thing is that Britannia was underpopulated, and that most of polities were shattered at this point in the island : even keeping the army in Britain wouldn't really help at this point, except maybe preventing the main post-imperial ensemble to shatter too much.
Note that the Saxon presence in Britain and Gaul feed from each other and not one against the other : more Saxons in Gaul because reasons mean more possibilities of Saxon settlement in southern Britain.