Getting baby two passports?

Hi there, just joined the forum and would like to say thank you to everyone for the great info provided here!

     I have a question regarding my soon to be born baby boy/girl (don't know yet!).  Our baby is due any day now and my UK wife and I (US) will be moving to the UK three months after the birth.
    The questions are, do I apply for SSN at hospital in order to apply for US passport as soon as we get the SSN back?  Is the SSN nessessary for the infant Passport?(I think the SSN is the slowest item to arrive)  Also I have read here that we could apply for the UK passport and avoid the need for a visa in the US passport.   I wonder how long the UK passport process takes?  Could all of this be done simultaneously and within three months of birth we might concievably have US and UK passports for our little one?
     Has anybody here done just this?  I would love to hear about experiences very similar to this, I wasn't able to find exactly this in my reading but would think it pretty common on this forum.

Other facts:

Wife is UK citizen with US green card(obtained through work)
We married in UK
Live in Denver Colorado currently
Moving to UK in early August
I am US citizen (Spousal visa for me sounds pretty easy from reading other posts)
I hope to become a Dual national after emigrating

The SSN isn't necessary for the US passport.  When my daughter's was issued, she got her SSN at the same time.

Your child needs a UK passport to enter the UK.

You will be eligible to apply for naturalisation 11 months after receiving ILR.

The passport(US) application form has a field for the SSN...this made me think it was necessary to get it first.  You say you received the SSN with the Passport so I guess you didn't request one upon birth, you instead applied for both sometime afterward say at a passport office?  How long did it take to obtain both?  Did you also get the UK passport and how long was that process?  Thanks for your help!

The passport(US) application form has a field for the SSN...this made me think it was necessary to get it first.  You say you received the SSN with the Passport so I guess you didn't request one upon birth, you instead applied for both sometime afterward say at a passport office?  How long did it take to obtain both?  Did you also get the UK passport and how long was that process?  Thanks for your help!

I applied for both at birth - well, at two months b/c that's the soonest I could get an appointment at the Consulate here after the birth.

Things are probably different in your case, however, b/c I had to register the births from the UK at the US Consulate.