How long after a hair transplant can you bathe in a hot bath (head under water)?

It is best to ask your surgeon about post operative care as each doctor has their unique regimen.   In general you can start washing your hair the next day.  

Hair
transplants come in two forms, a follicular unit extraction (FUE) and a
follicular unit transplant in strip form (FUT). 
For an FUE most of the scabs are closed after the 4th day, so
by the 5th day you can resume most activities. 
For strenuous activities I would wait a full week.     For an FUT (strip procedure), you would
need more time to allow the wound to heal. 
I would not bath with head submerged for a full week.  Staples are removed at 10 to 14 days.

Assuming that you don't stick your head under the hot water, you can take a hot tub batch the very next day. Don't soak the hair grafts for 2-3 weeks

I always recommend taking a week to 10 days before allowing direct water pressure on to the recipient area where the hair has been transplanted. For the most part you should be ok as soon as 48 hrs, but it doesn't hurt to just be a little cautious and give the grafts enough time to make sure that you don't do anything to disturb them. So take 7-10 days of being extra cautious and then after that you can submerge your head and allow direct contact from the shower to the area again.