The IRS will match your social security number and name on file with the Social Security Administration. It would be best to wait until your return has been approved and you've received your refund or paid your tax due to begin the name change process. Once accepted, it can take anywhere from a few days to 3 weeks to go from acceptance to approval.
Tip: It sounds like you may also be filing jointly for the first time. It's not important whose name goes first – at least not to the government – but what does matter is being consistent with each subsequent return you file.
When you file your first tax return together, decide whose name should go first and then stick with it per the IRS 1040 instructions, which read:
"If you filed a joint return for 2016 and you are filing a joint return for 2017 with the same spouse, be sure to enter your names and SSNs in the same order as on your 2016 return."
Changing the established name order isn't the end of the world, but it could result in processing delays for your taxes.