There is a tailoring material shop in Nadco in Andheri(west)..sorry forgot the name....they sell all the material required for tailoring, embroidery and also for making handicrafts. Or else you can find the threads at any decent fancy stores..especially where you get the material for school projects.
I don't know in Andheri, but at Dadar there is a shop called Mahila Vastu Bhandar. It's off Ranade Road on Chabildas Road. You get a lot of sewing items there. There are 2-3 other shops on the same road which also sell a lot of sewing items.