I routinely walk along the beach at Hengistbury Head (approx 40 miles further east) and have seen cuttlefish bones there for several years running. So perhaps, yes, there is a healthy population off the coast here.
[Edit] Went down a rabbit hole and found [0]. In summary, in the UK, cuttlefish are classified as shellfish in terms of reporting fisheries catches. Catches of shellfish have increased in absolute terms over the last 80 years, from 32k tonnes to 116k tonnes in 2022, and relative to catches of other fish. This is partly due to diversification of the fishing industry, partly because there may be fewer restrictions on shellfish catches.
The Channel is one of the UK hotspots for shellfish catching.
The Marine Conservation Society [1] states that "There is not enough data about cuttlefish in the English channel, but there are indications that populations are too small and fishing pressure is too high. There are no appropriate management measures for this species, despite high landings and high market value. Most cuttlefish are caught by beam trawling in the English Channel. This has the potential to cause damage to the seabed. Trawls can also have high bycatch, potentially of vulnerable species."
[0] https://www.gov.uk/government/statistics/uk-sea-fisheries-an...
[1] https://www.mcsuk.org/goodfishguide/ratings/wild-capture/905...