You'll be ok! We entered the UK at the end of February with my then 3 month old son. He only had a US passport and as soon as we arrived, I applied for his UK passport (arrived in 6 days).
After we received his UK passport, we left the country and then when we returned he entered on that one.
Also, you don't need a certificate of birth abroad, just his actual birth certificate.
Glad to hear you made it to the UK!