(As I'm sure Louis knows,) a tremendous amount of ink has been spilt giving different explanations for why this [edit- the seeming lack of influence from Welsh on English] happened.
From debates over exactly what vocabulary in English is derived from Welsh in both 'standard' English and dialects (the French embracing Celtic derivations, the English establishment seemingly not particularly to do so) and debate on how much the underlying grammar and usage of English is derived from Welsh,
Through to the fact that language influence in cases like this differs tremendously, depending on things such as the socio economic status and numbers of people coming in compared to the inhabitants, the different power play, the way the original language was regarded, etc- the situation in Britain was different in many ways to the rest of the Roman Empire.
Through to the idea that an underlying language having more effect on the grammar of a language than on its vocabulary is not unusual [tangentially, you can see that in Wales, where the way English is spoken is affected more than its vocabulary. Different times and circumstances- but that's the point, really. Language shift doesn't always happen in exactly the same way.] or even have little effect at all- to the degree it is often difficult to tell any influence at all from an underlying previous language.
And the idea that toponyms are often carried over, when other parts of the language are not.
All these considered, I'm not sure it is much of an unsolvable puzzle- rather it is a question with many different, not necessarily conflicting, answers.