The barges were quite flimsy, though some would make it across. The problem was that the French estimated that in order to cross the Channel and not be hopelessly exposed, they needed to cross at night. Unfortunately for the French, this would create chaos, and they knew it. Boats would end up along probably as much as 25 or 30 miles of the English coast, and there would be a British response by sea as well as land. The simple problem is that after the French under Villeneuve failed to drag the Royal Navy off to get lost and leave themselves exposed, they knew that they couldn't dominate the Channel. So they pretty much planned for the Royal Navy to make an appearance, counting on the sheer mass of ships to flock past the British vessels. This means they were expecting to lose large numbers - probably most of which would flee home, rather than drown at sea. So you've got to take into account that if the French did manage to get across, it would be something akin to the parachute drops on D-Day - men scattered all around for miles, battalions taking days to reform, and a large degree of organisational failure. Also you have to remember that the UK doesn't have continental commitments in 1804/5 so they have a large number of regular regiments to face down this invasion, and a large amount of militia. You could feasibly get something like the Battle of Stamford Bridge, where skirmishes rage on the edge of the beachhead as units struggle to reform on the beaches and march towards the fighting, and the British respond in kind to disrupt the French, leading to a series of small-unit fights which grounds the invasion force virtually immediately.
Just some observations, anyway. I'm not trying to predict how it would go. If anything, I'd say that any competent admiral wouldn't have the nerve to give the order to cross the Channel, anyway.