It is true that a language is classified into a language family based on the grammatical structure, particularly the morphology and syntax, and not on the lexicon. I'd love to see this book you mention, just to see these differences being linked to Britannic Celtic. I say that because not as much is known about the language as it was spoken at that time, and unless there has been a sudden influx of data, I don't know that one can officially link any specific change in English to Britannic Celtic. We have the most data from Old Welsh (800-1200 CE), but Old Breton would be the closest to the Britannic Celtic languages of England spoken at the time of Rome and of the time of Germanic invaders. The problem is that we have just a few toponyms of Old Breton, which is not enough data to make such claims. What is more, the Romans were not interested in documenting the languages of the peoples that they conquered, so without that contemporary data I don't know that one could say that the language of the Celtics influenced the languages of the Germanic peoples who conquered and settled in England.
There is another reason why I am hesitant to say that the Britannic Celtic languages influenced the Germanic languages, thus creating Old English: are we talking phonological changes, or morphological and syntactic changes? It's not particularly common for a substratum language to influence the superstratum language, even in the Indo-European languages. There are some thoughts of phonological influence--think the Latin [f] > Iberian and Gascon [h] phenomenon--but very little on morphology or syntax. The one possible exception is the Balkan sprachbund, where there seem to be quite a few common traits among the languages of the Balkans that are not shared with the other languages of those families. However, many have refrained from saying outright that there is a substratum influence, simply because the history of these languages during Medieval times is somewhat unknown, especially for Rumanian.
In addition to having a West Germanic grammatical structure, there has been influence on Old English by Old Norse--both in lexical items and object pronouns. But the greatest changes came at the time of the Norman Conquest--although it seems that the changes were already underway before 1066 CE. All one has to do is compare the Old English in Beowulf to Chaucer's Canterbury Tales and note the grammatical changes, and not all of this is due to influence of the Norman French spoken in the courts of the day. Similar changes can be seen in the history of Dutch, English's closest sibling, which (as I recall) has not had much influence from French or other Romance languages.
There are several books, and many that have been published or updated in the last few years, that chronicle the history of the English language. Charles Barber's The English Language is one handbook that just got updated this year, and one that I would recommend. (There are others, but I'm not as familiar with them...perhaps there are others on here that are.)
Clearly Irish grammatical structures have influenced modern Irish English, just as African languages have influenced 'black' dialects in the Americas. I'm not a linguist, but I wonder rather about woods for trees. I have lived in areas subject to quite recent language change (West Shropshire, for instance) and I think there is very clear influence from 'substratum' languages. I doubt that Germanic 'peoples' conquered eastern Britain - I see it more as a mercenary revolt.
Incidentally, for what it's worth, there is a long - and to my mind adequate, article on 'Brittonicisms in English' in Wikepedia - though I can't say that is a place I usually look to for confirmation!