Dr_Acula has noted the main problem - it's not just serial data which goes from the PC to the PICAXE, but also a 'break signal' a continuous high which has to be passed on. Some USB-Serial cables do not do this, XBee doesn't, and I'm not sure that Bluetooth modules would.
If you can find Bluetooth modules which carry the signals exactly as the Programming Editor puts out then it should be plain sailing.
The alternative is to use trickery to turn the signal from the PC into something which can be carried purely by serial, which can be converted back to what's intended at the other end. With this, any serial link could be workable, IR, IrDA, 433MHz, XBee, Bluetooth. It would still need testing, as there may be timeouts which need to be met which some setups might not cater for.
Converting the Programming Editor download signals into pure RS232-style serial can be done. I've proven that running a 'helper application' on the same PC as the Programming Editor. With download to COM1, COM1 TX goes out to COM2 RX, COM2 TX out to the PIXAXE end, PICAXE Serial Out goes straight back to COM1 RX.
Ideally I'd create a Virtual Serial Port so the Programming Editor would simply choose the Helper's serial port so operation would be totally transparent to the PICAXE programmer.
At the PICAXE end there needs to be something which takes the serial sent from the PC and converts it back to what the PICAXE Serial In expects ( which is what the Programming Editor download signal original was - or close enough to work ). That's theoretically possible, using another PICAXE. It's also necessary to force the PICAXE being programmed to reset prior to download, or it may be stuck in a SERIN or INFRAIN and won't respond to a download, which makes wireless download rather moot.
I haven't done any actual work on the PICAXE side of things and there may be issues which I haven't considered yet. It's certainly not as easy or as simple as it may first appear. Getting it to work over a hard-wired serial link would be the first milestone, then it should be possible to just drop in a wireless link while keeping fingers crossed.