RAW, there are no explicit rules for the speed of carts by themselves, they simply use the speed of whatever animal is pulling them.
For example, if a mule (40ft speed) is pulling a cart, the cart will generally also move at a speed of 40 feet. The relevant rules on vehicles are all found in the PHB on pages 155-157.
However, bear in mind that exceeding the carrying capacity will reduce your speed.
Notably, the rules on the maximum weight pullable by a draft animal differ from the regular carrying capacity rules. See PHB, p. 155 (emphasis mine):
An animal pulling a carriage, cart, chariot, sled, or wagon can move weight up to five times its base carrying capacity, including the weight of the vehicle. If multiple animals pull the same vehicle, they can add their carrying capacity together.
For example, let's say you have one pony pulling a carriage. The pony has 225 lbs carrying capacity, while the carriage itself weighs 600 lbs. Five times 225 is 1125 lbs, or 525 once you subtract the weight of the carriage.
Assuming that you load more than 525 lbs on the carriage (whether in the form of passengers or wares), the speed of the pony will drop to 5 feet, as per PHB p. 176:
Push, Drag, or Lift. You can push, drag, or lift a weight in pounds up to twice your carrying capacity (or 30 times your Strength score). While pushing or dragging weight in excess of your carrying capacity, your speed drops to 5 feet.
Note that the first sentence is overwritten by the more specific rules for carrying capacity of draft animals.
Either way, to summarize:
No, multiple animals don't affect the speed of a vehicle they are pulling, it will always move at the speed of the slowest animal pulling it (this is not explicitly stated, but obvious).
An exception occurs when one animal would have an insufficient carrying capacity. In this case, multiple animals would indeed affect the speed of the vehicle, but only insofar as it isn't slowed to a speed of 5 feet.