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.

Already-powerful high DPR builds get better defenses

RAW, if you combine Crossbow Expert, Sharpshooter, and the Archery fighting style, you have the basis for some of the most damaging builds in the game. This is (slightly) counterbalanced by the inability to use a shield.

With your proposed changes, those high DPR builds can wear a shield for similar defenses to a sword+board character. A plate-wearing sword+board fighter with the Dueling fighting style will have AC 20. A studded leather-wearing crossbow+board fighter with the Archery fighting style will now have AC 17-19 (depending on their Dexterity), instead of the AC 15-17 they would have without a shield. Dexterity is already a more versatile stat than Strength, and this will increase its power even more.

In case it isn't clear, the real problem here is the Sharpshooter feat. With it, the hand crossbow's normal range extends to 120 feet and you can take -5 to hit (-3 if you consider the Archery fighting style) to get +10 to damage. Combine it with the bonus action attack from Crossbow Expert and you get a terrifying amount of damage. If you restrict one-handed loading to be incompatible with Sharpshooter, most of the problems go away.

Implementation-specific considerations

1. "Tweaking the Ammunition property of hand crossbows specifically." This does not have any particular problems besides the hand crossbow + shield ones listed above. It does allow for niche shortsword + hand crossbow fighting, which is flavorful for a Swashbuckler rogue, but not particularly powerful.
2. "Adding a feature to the Crossbow Expert feat." This also doesn't have any particular problems, either. Shortsword + hand crossbow characters would need to take Crossbow Expert to work.
3. "Solve it through magic by making the fancy crossbows of the royal guard uncommon magical weapons that can be reloaded without having a free hand." This is essentially the same as point 1 once the PCs get their hands on one or two of the royal guard crossbows. Depending on your players, this may also encourage them to...relieve...one of the guards of their crossbow. On the other hand, Ben Barden rightly pointed out in the comments that forcing the players to choose between a +0 repeating hand crossbow and a +3 hand crossbow may give you greater control over balance.

Cost/benefit for exploiting one-handed loading

Fighter (and most other martials)

There is definitely a cost to taking Crossbow Expert and Sharpshooter, but in this case, the cost was already "paid" once you chose to play a ranged DPR build. If you want to play optimally, you're going to want both of those feats anyway. Here are four different ranged fighters (using no subclass features) at level 6, all with the Archery fighting style, shooting at an AC 15 enemy:

• +2 DEX at L4 and L6; 2 longbow attacks for 15.7 DPR.
• Crossbow Expert at L4, +2 DEX at L6; 3 hand crossbow attacks for 17.4 DPR.
• Sharpshooter at L4, +2 DEX at L6; 2 longbow attacks for 19.0 DPR.
• Crossbow Expert at L4, Sharpshooter at L6; 3 hand crossbow attacks for 22.8 DPR.

View the AnyDice program that generated these

Keep in mind, this is the worst level for Crossbow Expert + Sharpshooter, and it's still on top. At level 8, the first fighter already has 20 DEX and can't go higher, but the Crossbow Expert + Sharpshooter fighter gets even better. Any bonus to hit (from +X weapons, proficiency, bless, or Battle Master's Precision Attack) only widens this gap.

Rogue

The key to a successful rogue build is landing Sneak Attack every turn. The standard way to accomplish this is to attempt to Hide each round to gain advantage on your single attack. Unfortunately, hiding is sometimes impossible, leaving even a high-level rogue with a single d20 roll to determine all of their damage for the round.

The Crossbow Expert feat provides an alternative, by offering a different use for the rogue's bonus action: another attack. Two attacks (without advantage or disadvantage) are strictly better than one attack with advantage, making Crossbow Expert's bonus action better than Cunning Action in most cases. They may both hit and maybe even crit. Better yet, if the rogue connects with the first attack, they can still choose to use their bonus action to Hide or reposition.

Changing the Crossbow Expert feat won't have a huge effect on pure rogues, but may encourage them to dip a single level of fighter for shield proficiency and the Archery fighting style. Tweaking the Ammunition property itself would mean that rogues with a fighter dip wouldn't need to take Crossbow Expert if they're okay with a single attack.