Sailing 'up' the channel (as the Spanish Armada had discovered) was risky, not least because of the bad weather that was present as the dispatches arrived on the English coast.
The vessels (HMS Pickle and HMS Entreprenante) that were given the dispatches were small and relatively fast but neither was powerfully armed. The reason that duplicate dispatches were sent on different ships was that these vessels could be lost, because of the natural risks at sea, or intercepted by the enemy. While the British Navy had the main French fleets bottled up in their ports, there were still numerous French cruisers (both regular navy and privateers) waiting to dart out into the channel to pick off any British ships that they could. A small, lightly armed dispatch boat would have been a tempting target.
Pickle's commander was John Richards Lapenotiere. His orders were to land the despatches at Plymouth, which would still have involved a considerable overland journey to London, but the failing winds at the time prevented that:
Meanwhile the wind was dropping and Lapenotiere set every stitch of canvas including the studding sails and ordered out sweeps to keep the Pickle’s head pointing towards England. He was clearly not going to make Plymouth but presciently his orders stated that ‘should you be prevented by Easterly Winds from fetching so high up as Plymouth, you are to make the first port you can in England’. So, off Falmouth on 4 November 1805, Almy’s log reads, ‘at 9.45 Shortened sail and hove to & out Boat our Commander landed at Falmouth with his Dispatches’.
Mariner's Mirror Vol.91 No.2 (May 2005)
By landing the communications at Falmouth, they could be sent to London without any further risk of being lost at sea, either because of the weather or enemy action. Falmouth was also home to the Post Office Packet Service, so the route to London was well established and the harbour was well suited to despatch vessels.
John Richards Lapenotiere and H.M. Schooner Pickle and Their Fifteen Minutes of Fame, Peter Hore, Mariner's Mirror Vol.91 No.2 (May 2005), 284-293