How long does it normally take the IRS to process a 1040X form?

There's no set limit to how long the IRS takes. If you file early they'll be less busy and usually it will be done in three weeks. However if you want to the last minute it might take twice as long.

If you file your 1040X during tax season you can expect it to take up to 12 weeks to be processed, however there have been reports of it taking up to 18 weeks or longer. During the off season it appears that it takes 6-8 weeks on average.