To my knowledge, Christianity had already established itself to some extent within the British Isles during the Roman period and was already the predominant religion amongst the Romano-British elite and urban population. Was it simply that the Christian presence, unlike in areas such as Gaul and Italia that were settled by the Franks and Longobards respectively, was so minimal that it was effectively overwhelmed by the influx of pagan Anglo-Saxons and vanished into obscurity before it could leave any real impact on them?
Well it's a bit of an open question. Christianity was certainly present in Britannia by the end of Roman authority in the west, and it not altogether vanish despite the influx of pagan Angles, Saxons, and so on. Christianity after all did survive in Britain, the Anglo-Saxons did not totally depose the native Britons and their religion, but it certainly went into a state of decline. Now why might this have been?
Robin Fleming in Britain After Rome posits that due to the economic collapse in Britain at the end of Roman presence there that British religious life withered on the vine in the absence of Roman infrastructure and city life. Without the city life of Rome keeping it afloat the various dioceses of Britain were essentially rudderless and powerless to resist the newcomers. On top of this, due to Anglo-Saxon dominance in much of the old roman province, continued adherence to Christianity was disadvantageous in the long run and the native inhabitants assimilated to their new pagan neighbors and rulers rather than the other way around because of their economic and political subservience.
Without the political and economic, not to mention military, support of the Roman empire Christianity was very vulnerable in Britain due to its collapsed economy and lack of centralized rule capable of resisting the pagan newcomers.
So how is this different from the rest of the Roman west? For one Britain, and northern Gaul, were far and away the most economically declining regions of the empire due to the increased regionalization and de-specialization of the Roman economy. Other areas such as Italy a d southern Gaul were more economically secure and able to sustain Roman institutions far better than Britain was able to. In many of these other areas local Roman notables were able to maintain some sort of political and legal autonomy. Perhaps most importantly though, the Lombards, Ostrogoths, Visigoths, and so on were already Christianized before they settled into their various kingdoms. While often adhering to Arianism, deemed heretical by the mainstream church, they were still Christians.