If you make a brand new stone stockpile, the number of wheelbarrows assigned to this stockpile will be 1, because that's the default. (For any other kind of stockpile, the default is 0.)
You can see how many wheelbarrows are assigned to a stockpile, or change the number, by doing a q(uery) on the stockpile.
When a stockpile has a wheelbarrow in it (or when the wheelbarrow is in use), all hauling jobs to that stockpile must use the wheelbarrow(s).
So, let's say you make a brand new 10x10 stone stockpile and don't change any settings. Then you unpause the game.
At that moment, up to 100 "bring stone to stockpile" jobs can be created, and you could have up to 100 dwarves hauling stones on their backs at the walking rate of a wounded tortoise. In addition, one "bring wheelbarrow to stockpile" job will be created.
Let's suppose that you don't have 100 idle dwarves at that time. Maybe you've only got about 10. So, 1 dwarf brings a wheelbarrow to the stockpile, and 9 more dwarves are still carrying rocks on their backs.
Once that wheelbarrow is actually in the stockpile, then no further "bring stone to stockpile" jobs will be created. Instead, one "bring stone to stockpile using this wheelbarrow" job at a time will be created. One dwarf will go to the stockpile, get the wheelbarrow, then go to the nearest qualifying stone, put it into the wheelbarrow, and return to the stockpile. All of this will happen at normal walking speed.
If you're clever, you can alter the stockpile settings before you unpause the game, to prevent the "bring stone to stockpile" jobs from being created before the wheelbarrow is in place. What I like to do is assign the desired number of wheelbarrows and disable all stone types until the wheelbarrow(s) are there; then turn on just the specific kind(s) of stones I want in that stockpile.