Though I don't bother cleaning my uilleann pipes' wooden parts (they don't seem to need it), whenever I oil my wooden flutes' bores, I also wipe down the outsides with plain old bore oil. Sometimes I remove the keys first; often I instead wrap aluminum foil over the pads, and oil around them. After letting the oil sink in and dry overnight (much of the excess evaporates), I wipe the outside with a nice soft cloth.
This is all standard procedure for wooden flutes, which are made of the same hardwoods as pipes, so it should be equally applicable. Bore oil is intended mainly to keep the wood supple enough to expand & contract through the seasons without cracking (which, actually, is seldom an issue for bellows-blown pipes), but it also loosens up surface grime, and my flutes look terrific when I'm done. So, bore oil is probably also a good choice for cleaning your pipes.
People sometimes argue the merits of bore oil, which I think is usually petroleum based (and therefore arguably not terrific for lip contact), versus almond oil, which some people fear can turn rancid unless vitamin E is added. But I think the rancidity issue only matters with stored almond oil -- you shouldn't need to worry about almond oil turning rancid on the instrument. Either way, Mr. Loos' advice is certainly quite trustworthy.

Regards,
Mick