Further this is my suggested fool proof way of doing it:
1. Apply with ITIN. Wait until you get your refund or money deducted from your account if you have to pay tax extra, so you know your tax return has been processed. Only after this go over to local IRS office to inform of SSN to merge records.
2. Get your SSN, go over to local IRS office, do the thing, and then after a week (just gut feel, give them a week), file your tax returns with wife's SSN.
Generally once a timing issue or a deadlock or something slips through cracks is created in a government process, a lot of heartache is taken to resolve it, want to avoid that.
I had done number 1 above as I couldn't wait to get my refund, wife didn't have EAD in hand till then, and guessed it will take like at least one to two weeks after EAD to get SSN
