I can't see most of modern day France but if you consider burgandy part of France (in the most decentralized way)
https://en.m.wikipedia.org/wiki/Burgundian_State
I could see a Anglo-Saxon England cooperation with the HRE to enforce thier claims on central burgandy while England focuses on seizing the burgundian low country's for its self due to its easy access to the english logistics network just a stones throw away from the channel. If england goes down this route the Low Franconian areas are the easiest areas for England to access and the most likly to fall into at least its sphere.
en.m.wikipedia.org
They would also probably push into otl french Nord Pas de Calais in order to control the eastern entrance of the english channel
en.m.wikipedia.org
Linguisticly the germanic parts of the low contrys modern day are closer to otl english then both French and German likly do to the german dialect english developed from being being ajacent to the low country's dialects through Old Saxony
en.m.wikipedia.org
so without otl French influence on english in this timeline long term english ownership of the low countries should make english and the low contrys even more similar linguisticly. Interestingly while a saxon England does not have any claim in France like a Norman England they have theoreticl claim to the saxon HRE
en.m.wikipedia.org
if the HRE and England maintains good relations I see England allying with thier fellow Saxons and building out a sphear of influance but if HRE turns hostile at some point (prehapse wanting to eject Anglo saxon England from the neatherlands, or feeling threatened by Englandbuilding its own sphere of influance) I could see England trying to forge a alliance with the Scandinavians or France (assuming England did not go further beyond Nord Pas De Calais and piss them off) to keep them off. If england comes out on top long term I could see them consolidating thier influance in the old duchy of Saxony and prehapse in a few centuries try to unite them in a crown in personal union with England kind of like otl Hanover.
en.m.wikipedia.org
(In short Anglo Saxon England is likly to have limited stratigic reasions to invade france [control of the channel] but if we extend the idea of France to include burgandy and thus the low country's there are more things to motivate England [similer germanic dialect, the high value of medieval flanders cannot be understated and control of the mouth of the Rhine gives them control of a huge economic artery] beyond the idea of the "natural" french border of the Rhine reasions for english intrest start to drop off but if are still present you are willing to streach the idea of france to its maximum limmit through the frankish empire which allow you to consider the old saxon region which would be more meaningful to a anglo-saxon england then it would to a norman England)
en.m.wikipedia.org
en.m.wikipedia.org