I had DD in the historic 2008 Adelaide March heatwave where it was 15 days straight of over 35 degrees. I think she was born on day 5 of the heatwave!
She wore nappies and singlets and also body suits now and then (you know, the legless ones). I didn't wrap her during the heat wave, but at night would just put a muslin over the top of her and tuck it under the mattress.