It depends a lot on your dating of the disappearance of Gaulish and Celtiberian (I am going to use Gaulish as an example, for obvious reasons; hopefully, someone will be able to complement it by information on the Spanish case). If you think that Gaulish was long dead when the Franks, Visigoths and Burgondians (to centre on the most important players in this area) took over, well, it is rather difficult to explain. While relatively far from the centres of power, parts of Britain certainly were more Romanised than, for instance, Northern Gaul, and Latin may have subsisted as a living language there at least until the 6th century. On the other hand, if you are ready to accept the idea that Gaulish was still spoken (something that I was unwilling to believe a few months ago, but I tend to become less and less skeptical of it) in the 6th century, things become easier to explain: post-Roman politics were the exclusive cause of this difference.

Frankish settlement and influence was maximal in Northern Gaul, the less Romanised part of the Roman province (and the most likely to have been home to a substantial Gaulish speaking population). On the other hand, Anglo-Saxon settlement centred on the most Romanised part of Britain, wreaking havoc amongst the people who would have been the most likely to develop a Brittonic Romance language. Conversely, northern Britain, the highlands of Wales, and Cornwall, marginal regions where native languages still were spoken, suddenly became the centre of native polities, and certainly of their chiefs. Britons naturally adopted this language (a fortiori since the prestige of Latin learning must have faltered after the desertion of the Romans). Meanwhile, southern France remained a bastion of Romanitas, and the Frankish aristocracy itself probably adopted the Latin language quite quickly, probably because of the contact with this lively sub-Roman world. As a result, the conquest of Latin (or proto-French) was allowed to continue, because it had become (remained) the language of power.

(as a side note, developments similar to the Welsh situation can also be observed in Gaul and Hispania, in the Basque country: a native language survived Roman rule in the highlands and its speakers formed dynamic polities in the post-Roman era)