There is an Australian woman who does kids parties, I think her place is called Fantasy parties. Although I don't know if she does them for kids as young as 1, they tend to be a bit older and more into playing games etc - you'd have to ring and ask her.
I recently did a google for birthday parties in Singapore and came up with this website which you may find useful, I think Fantasy parties are actually listed on there too:
http://www.singaporeforkids.com/kidscor ... arties.php
I'm organising my daughter's myself this year as she wants to invite the whole class. I'm actually dreading it - having a bunch of 8 and 9 year old kids running around going crazy........nightmare, but it's what she wants and I can't afford planned party rates.
I'll be needing a few stiff drinks after that particular afternoon!