Very unlikely, and very dangerous. The Channel is the busiest shipping lane around Britain, and can be one of the roughest, with regard to the weather. The prevailing winds are westerlies, so the weather travels up the channel, being squeezed, all the time, by the narrowing "funnel." Remember what happened to the 16th century Spanish Armada, and the D-day Mulberry harbours in June 1944. A few years ago, an attempt was made to find Glenn Miller's aircraft, after the position was fairly well pinpointed, because there was (allegedly) a briefcase containing some unplayed music. All they found was an engine, which might have come from the type of aircraft, but everything else had corroded and vanished without trace. Aluminium doesn't survive well in salt water, especially when it's continually on the move.
As the reappearance of the Spitfire and Dornier show, the shifting sands will sometimes reveal items, but can just as easily bury them. Further round the coast, things do still appear; during a 1980s visit to Binbrook, to view the Lightnings, our guide told us how they were often dropping aircraft into the North Sea, and fishermen were always claiming for damaged nets. He told, though, how one claim was refused, since they were able to point out that Lightnings had never carried black crosses on their wings, and the suggestion was made that they should enter their claim with the authorities in Berlin.