SSN name change/Registering birth abroad

I've just noticed the note on the US Embassy website that says "It is important to note that your names as on file with the Social Security Administration must match the names used on the application for a number for your child."

When I got married I was already living in the UK and to be honest, never even considered changing my name on my SS card. Is this going to cause problems? Should I delay registering his birth to sort this out? Has anyone else done this without having changed their name?

It is probably worth noting that all of my documents here are in my married name.

Is the name on your passport different than that on your Social Security card?

You can sort it at the same time, I know someone did it recently

So, I sent a quick email to the embassy to ask them about the SS card and they said that the application for the girls cards asks for my maiden name in any event, so this isn't a problem. So I'll be able to get theirs no problem and get mine sorted in my own time.

Evidently it's not a problem!

It's not a problem.  I had the same worry as I had renewed my passport in my married name, but not got around to changing my SS number.  I was able to get little one's stuff with no worries.  I don't know why they say that.  It must mean something different than it seems to.

By the way, you can do it while your there for your child.  I just had to go to a different window and show them my marriage certificate or something like that.  I didn't have an appointment, I just asked someone while I was there and they gave me instructions, so now all my info is up to date.