Hi there:
I'm happy to add my two cents while enduring pregnancy-related insomnia...

There are a lot of baby/toddler-oriented stores in Singapore -- nothing as comprehensive as Baby Bunting in Oz or Babies R Us in the U.S., but you can still find a lot of products (including Baby Bjorn & Nuk) here.
The best stores I've seen -- in terms of western product lines -- are Motherworks on the ground floor of Great World City Mall (you can google for specific location of mall); The Early Years on the 5th floor of the Paragon Shopping Center on Orchard Road; and Mums & Babes on the first floor of United Square Shopping Center in Novena (Thomson Road). United Square is one of several malls that caters to the "kiddie set" (an entire section of it is devoted to children's clothing, toy & gear stores).
In a pinch, if you can't find something here, it's not uncommon for people to ask friends/family back home to send (particularly if not bulky). Additionally, my friends here have been great about saying "We're going to the U.S./Oz/U.K., is there anything we can bring back for you?" I am happy to do the same for them when I go home to the States.
Good luck with your upcoming move!