I wondered the same thing about actually getting the registration/birth certificate from the High Commission.
The advice they gave me was that the advantage of registering the birth there (regardless of whether you got a copy of the certificate (you don't have to, and it costs $98 if you do)) was that when the child returns to the UK, and need a replacement birth certificate (well, who doesn't lose things) they can apply from the NSO in the UK. Otherwise, you'd have to get hold of the government office (or possibly the hospital) here in Singapore to get a copy. Bit of a pain if you needed it in a rush.
We chose to get all done (including copy), that is passport and registration, and it cost us $337.80 (I believe).