I'm very interested in the possible influence of Phoenician, specifically, on local dialects in the British Isles during the Iron Age. I'm curious about any historical and linguistic evidence that may exist, such as inscriptions or loanwords, and how it may have affected the phonology and grammar of the dialects. I would appreciate any information or resources that could help shed light on this topic beyond Theo Vennemann and linguistic coincidences. I'm aware that there is a broader discussion on North Africa and the British Isles, but I'm interested specifically in Phoenician. Thank you, as always, for sharing your time and expertise.
There is no such credible evidence.
The closest we get is some archaeological evidence of trade routes between Carthage and the Southern British coast (from what I remember this is mostly in the form of British tin being found in and around Carthage), but certainly not of the kind of consistent settlement you'd need for significant linguistic effects.
It's worth noting as well that even in Southern Spain and the coast of North Africa, where we do have good evidence of significant Punic populations, the linguistic influence is almost exclusively limited to place names (e.g. Cadiz, Malaga, & Cartagena in Spain).
As detailed in my answer to the linked question on "unexplained similarities" between (Insular) Celtic languages and those of North Africa, Vennemann's claims are spurious. These features are not especially unusual cross-linguistically, so there is no need to invoke contact to explain them, the timeline of the development of the features in Insular Celtic rules out contact with North African languages as a cause, and in many cases the "similarity" of the feature is greatly exaggerated.