In this specific case I wish to connect my first computer wirelessly to a router, and have the laptop use a bridged connection to Ethernet to extend to other computers.
The other two computers will be connected to a layer 2 switch, and the switch plugged in to the Ethernet port of the first computer with the bridged network.
Would this allow the two other computers on the switch to access the internet?
Sure, that will work just fine, and in fact I once set up a couple of macs in exactly the manner you describe. You could either set it up as a true bridge (layer 2, all machines appear on same network) or as a router (layer 3, machine behind the wireless one is on a separate network). A bridge is simplest in this case.
Be aware of course that the computer in the middle has to be left running all the time for this to work consistently. Once you turn off the machine with the wireless connection, the system behind it will lose connectivity.
A better way to do this is to set up an inexpensive wireless bridge instead. For example, I used a $30 Asus wireless router running dd-wrt to bridge my directv receiver to my home wireless network. This approach is so inexpensive these days that it seems the better way to go as opposed to making a regular computer system act as a bridge or router.
