Will have 3 days in NO in June. We will not have a car.
Are there any recommended companies that combine a swamp tour (just have to see those wild gators) and a plantation tour in the one day ?

Ooooo, that does sound like fun. I sure can't help but hope you manage it.
Some ways west of NO is a place on the Interstate where one drives for around 15 or 20 miles on a bridge over a large swamp. I am really fascinated when I look out at that gigantic body of water. Once we were driving around 60 mph (just a little faster than the low speed limit on the bridge) and out on the water, we could see a boat keeping up with us. I joked must be drug runners...

Google the words New Orleans swamp tours and you'll get a ton of listings. This is the official tourist website. I would just go through the listings until you find a company which offers the kind of tour you want. I remember that there are several companies which do a one-day combo tour. And most offer FQ hotel pick-ups.

I had a rental vehicle so arranged a trip to the plantations and the bayou on my own. However, as #3 says, the "swamp and plantation" combination is a common one, so you should be able to find plenty on-line, or you can ask your hotel or a tour kiosk in New Orleans.
Gray Line Tours seemed to be a big company; they run tours to Oak Alley Plantation (one of the "classic" Gone with the Wind type plantations) and a bayou tour in one day. Gray Line Tours