This is an example of a type of British humour that will not be easily understood by a lot of people - the "joke" contains irony. Of-course William will have no fear of getting home because he has car to take him and doesn't use the tube. By pretending to get up and leave quickly he is reflecting (in a friendly, comradeship kind of way) what other people would be feeling like doing on hearing the announcement. The "joke" is subtle and no words needed, but it is why others (most likely who would have used the tube had there not been a strike) laughed too. William wasn't saying "hahaha, I taking the **** out of all you tube users", come one, he would't do that.
A joke I once made fell completely flat because the woman I made it too wasn't British. It was a horrendously wet and windy day and I was walking into a hotel lobby in the US and a woman was standing just inside the door wondering whether to go outside or not. I said in a humorous way, "it's a little bit damp outside" and she said "what are you talking about, it's pouring with rain" and she stomped off.
Londoners are a tough bunch of people, for many, they will either work from home, work bit later and wait for when the buses or taxi's are less busy (my partner did that and had a nice meal out), stay over a friends or whatever. The tube is notoriously horrible to travel on at rush hour, if anything some people will have been relieved to have needed to find a different way home.