Kensington Nursery at DSO is brand new - not even open yet - so contact their Umm Sequiem branch. They do accept babies in their Baby Haven section. It's quite expensive though....Due to open 11th Sept.
http://www.thekensingtonnursery.com/
This will be opening in the Cedre Villas Centre in September. I think they do take babies as young as yours as far as I'm aware.
I am not sure there is one. also most Nurseries do not except kids below 2 years.....
If nothing comes up you can check the ones in Int. City, it's very close + they have transport most of the times.
Kids World is the best there. They have a branch in JLT too :)
Good luck :)