Look at this website
for St. Thomas beach information.
Maho is a good 10 miles from where the ship docks, and given the traffic on St. Maarten it can eat up 30-45 minutes each way. If you want to try it, a taxi is really the best choice.
You really can't hit all three beaches on St. Martin. The transit time makes it impossible given a cruise ship schedule, and bear in mind that if the itinerary says the ship leaves at 5pm that means they are pulling away at 5pm. They want the passengers back aboard at least 30 minutes earlier (4:30pm). This is true for all ports.
The beaches at St. Maarten/St. Martin can have some heavy duty tides so please be careful with your child in the water. There is a public beach right where the water taxi from the ship's pier to the main downtown is located. Yes, even though the cruise ship will dock the dock is off a little ways from Philipsburg they have a water taxi (it costs a few dollars for an all-day pass) that cuts down on the time needed to get to the town. You can walk it, but it takes a while and is through an industrial/seedy area.
Your plan for Nassau is fine. There is a free beach very near Atlantis that is nice. Atlantis is not a cheap place for anything.
St. Maarten is really the best choice between the two for purchases of electronics.