[RPG] Can a ballista be fired once per round with a crew of three

You load as part of any attack with a crossbow. You can draw the ammunition as part of the attack, so loading it is implied to be part of the attack as well. "Loaded" in this case means only that you have ammunition available for the attack. Per the official errata, "loading a one-handed weapon requires a free hand."

This means that, post-errata, using rules-as-written (RAW), you cannot indefinitely fire two hand crossbows in the same turn. If you have a single attack, you can fire once with your action and once with the second crossbow as your bonus action with Crossbow Expert on the first turn of combat if they start loaded. However, you can't keep dual-wielding because you won't be able to reload with both hands full.

If you stow the second crossbow on the first turn with your free object interaction, you can fire the main crossbow twice on the second turn, since Crossbow Expert does not say the bonus hand crossbow needs to be different than the one-handed weapon that triggers its bonus attack. You may need to load it before the bonus attack using your object interaction, depending on your GM's interpretation of the feat. You can do this on each subsequent turn to fire a single hand crossbow twice per turn (or more than that, if you have Extra Attack).

You didn't request a rules-as-written answer, so I'll note that there is nothing game-breaking about allowing dual-wielding hand crossbows using Crossbow Expert.

Because of the way crossbows work, I think your GM would be perfectly justified in waiving the errata rule in this case but requiring you to get a customized crossbow and ammo belt to allow you to load it one-handed. That would be part of your house rules, though.

There are 2 definitions of "round" in 5e. There is the definition of "round" that you quote, which is from initiative position intMax to initiative position 0 (or negative if you manage that some how).

Then there is the definition used here for readied actions. This is the same definition used in "once per round" effects such as certain powers. This definition begins at the beginning of your turn and ends at the beginning of your next turn.

Because (unlike in 4e), readying an action and activating it, doesn't move your initiative order position, and because it matches the readied action refresh timer (which happens to use the same 1/round definition of round), readied actions can be used at any point in either the current round or the next round prior to your turn.

Here's how reactions refresh:

When you take a reaction, you can’t take another one until the start of your next turn. If the reaction interrupts another creature’s turn, that creature can continue its turn right after the reaction. (Player's Basic p70)

This is a large part of the support that leads me (and many others) to believe that readied actions can roll up to your next turn.