Yes, we call them ice cream vans. I'm not sure of the specific song either but it's something that you always just know when you hear it. Whenever you hear it you know what's coming. You could just describe it as the familiar tune.
As for specific ice creams. Most vans will sell Mr Whippy in cones and you can get single or double cones, some might do waffle cones, you can get it in a little tubs or also in plastic cone-shaped things which have a ball of bubblegum at the bottom and are called screwballs. You can usually add a flake or some sort of sauce, strawberry, chocolate etc. I used to call a cone with a flake a 99, I think because they used to cost 99p but don't know if that's still used.
Then in addition to the ice creams you get lollies.
Some examples are:
Fruit Pastille Lolly
Calypo (not sure if I've spelt that right)
Some sort of plain fruit lolly - orange/strawberry
Oh wow, I can't actually think of any more. I guess I'm too set in my ways for what I usually have.
Also, ice cream vans sometimes then have a selection of drinks and sweets.
Hope this helps