It seems everyone is pretty much ignoring the OP-from the title "Britain is settled by Franks instead of Anglo-Saxons." And glancing over the facts (which seem to be pretty murky actually) it seems there is good reason for this.
For perspective:
1) Rome ordered the last legion out of Britain in 410. The intention was I believe for them to return pretty soon, but that never happened.
2) By 500, OTL, a diverse group of probably not too well organized Germanic peoples we might as well loosely call "Saxons" were in effective occupation of at least the eastern half of modern England.
That's our time window, and to totally preempt the "Saxons," or anyway limit their incursion to small foederati groups held to the coasts and prevented from the wholesale overturn of the whole island south of Hadrian's Wall, we'd better have the OP's imagined "Franks" preempting them with strong effective organization and political control all the way from Dover to Eboricum.
The OP assumes
"A Frankish Britain would have kept Nicene Christianism and maybe Celtic and British Latin dialects had survived for longer. "
If in fact a highly organized, large Frankish force with the intention of systematically replacing the entire Roman organization of the island, leaning as the mainland Frankish takeover in Gaul did on coopting surviving Roman institutions wholesale for their own profit, were available, and accepted by the Britons as the lesser evil, I'd not only endorse but amplify the latter clause; my expectation is such a Frankish Britannia would wind up some centuries down the line speaking a language fundamentally Britannic-Celtic in its grammar and commonest words, strongly influenced both by Germanic Frankish and by Latin--in fact the latter would have the greater influence and the language might be mistaken at a cursory hearing for a Romance language.
But I think the numbers are all wrong for this.
OTL for instance, the first mention I can find for any "Franks" adopting any Christian rite is the conversion of king Clovis, around 500. We can guess individual Franks here and there converted earlier than this, but there is no mention of any wholesale adoption of any brand of Christianity by them before this. 500 is far too late! These Franks on the Continent certainly did Romanize, but by the time they did, Britain is well on the way to becoming "England" as we know it.
In order to justify the OP's optimism a semi-Romanized band of Germanics with the intention of perpetuating (for their own use) Roman institutions including the established Church, rather than steamrollering right over it and setting themselves up as free Germanic bands instead, the best I can do is suggest a POD in the 4th century where some ATL Roman initiative attempts to both Romanize and Christianize the Salian Franks on their far north. But I think one would look in vain for a compelling reason for Roman authorities in the later 300s to try such a thing and it is an even longer shot they'd succeed!
The probable thing, or more probable, is that some "Franks" of the early 5th century indeed opportunistically invade Britain, with the same means, and same ends, as the OTL disparate bunch forming what are retrospectively called "Saxons" and which the Catholic Church of later generations collectively called "Angles." The upshot of that is one or two groups of Germanic invaders of OTL would be preempted by these Franks, and thus the melting pot of disparate but related groups is stirred a bit, with one dialect replacing one or two others. But fundamentally, the character of this ATL mix with Franks in it instead of someone else is going to be pretty similar! The upshot is going to be driving the native Britons, at least those who keep their language and cultural distinctness intact, west to "Wales" or to other refuges overseas, with non-Christian Germanic pagan kingdoms accreting into the Heptarchy more or less as OTL, for Irish Celtic Rite and continental Roman Rite missionaries to seek to Christianize anew. The ATL amalgam might or might not be called "England" and I suppose the linguistics of 'Old English' would be somewhat different, but of a similar character and content.
Actually the Wikipedia article on the Salian Franks mentions that there seems to be little distinction to make between "Franks" and "Saxons" in the 4th century--any meaningful distinction would boil down to the degree to which "Franks" have assimilated aspects of Roman civilization they intend to perpetuate.
I suppose it makes sense people are mainly talking about events in the time of Charlemagne then, but it makes little damn difference; by then the missionaries had done their work, England was collectively a Christian land, and any ATL claims of kinship to part of the crazy quilt forming England would be far fetched whims, offset by the realpolitik Charlemagne faced where a broad front of unsubdued continental Germanic pagans faced his Frankish realm, and he would prioritize that just as OTL, and cultivate the English as friends, not to muddy the waters with dubious lateral relationships putting an imperial foot in the island's door.
It is kind of cool to vaguely imagine WI some century-precocious gang of Franks all set to leapfrog right past the chaos of Merovingian Francia (straight into the chaos of post-Charles the Great Carolingian Imperial Francia, most likely!) were to 1) convert to Christianity, the Roman rite, a whole century before Clovis; 2) become Romanized enough, presumably as foederati, to appreciate the value of seizing control of some Romanized land wholesale; 3) seize the opportunity handed them by Rome pulling out the last legion in 410, perhaps also motivated by collectively being on the wrong side of some dynastic quarrel, perhaps the same one that yanked the Valeria Victrix out of Britain, and thinking, rather than face the music of a returning vengeful Emperor of the wrong faction, leap over the Channel to make the staggering Britons and remnant of Romanized persons there an offer they can't refuse.
If we Mary Sue the hell out of it, I can see the Britons rallying to such needed leadership, acclaiming the Frankish leader their king, the Franks (big Mary Sue Hail Mary Pass here!) revising their own customs and institutions to avoid the insane fragmentation of realms dividing them between the prior king's sons and instead planning a single unified succession, and this unified Frankish realm holds the line on both coasts--for even as "Saxons" were invading to lay the groundwork for "England" from the east, other invaders were raiding and settling from Caledonia and Hibernia; the Frankish unification of Romano-British potentials would have to face both fronts. And win.
Well, if they can do all that, I guess Britain as I said becomes a fundamentally Celtic-language remnant of the Classical order. Maybe such a kingdom can repel the Norsemen too. If it can endure 500 years it has to stand against a probable Norman conquest attempt.
Anyway, this would not be England at all, nor would it be known as a land of the "Franks," their cousins over the Channel have a lock on that.
It would be Britain.
Oh dear.
