I was going to suggest using Serial Line Internet Protocol (SLIP) as a means of getting UDP/IP data from a PICAXE to a PC, router, BifferBoard and similar. The PICAXE can build the packet up and just send it via serial, the PC OS will do all the rest, and it can even receive and pass packets back to the PICAXE via serial.
It's a very simple protocol, should require minimal configuration, and 'everything' supports it; Linux, Mac, Windows. Except it appears Windows 7 ;-}
It may be the best way to use a second-hand PC or laptop. With an ISA bus and a 3com NIC it should be possible to get it all working with MS-DOS, no Windows required. An old 8086 (!) or up should be enough.