My impression is that whether it's you or the familiar, the action economy is the same.
From Reload on PHB-279:
While all weapons need some amount of time to get into
position, many ranged weapons also need to be loaded and
reloaded. This entry indicates how many Interact actions
it takes to reload such weapons.
So the Interact action is specifically for loading the bolt into the crossbow and getting it ready to fire.
Now for the familiar:
Even if you are sharing a space, you and your familiar are two different creatures. Rules for drawing items are in Table 6-2 on PHB-273.
Pass an item to or take an item from a willing creature(2): Interact.
2: A creature must have a hand free for someone to pass an item to them, and they might then need to change their grip if they receive an item requiring two hands to wield or use.
This is a little weird because you would technically need to hand the bolt to the familiar, but the Advanced Players Guide fixed this with Valet (APG-147):
Valet: You can command your familiar to deliver you
items more efficiently. Your familiar doesn’t use its 2 actions
immediately upon your command. Instead, up to twice
before the end of your turn, you can have your familiar
Interact to retrieve an item of light or negligible Bulk you
are wearing and place it into one of your free hands. The
familiar can’t use this ability to retrieve stowed items. If the
familiar has a different number of actions, it can retrieve
one item for each action it has when commanded this way.
Valet specifically says "place it into one of your free hands" but sure a nice GM will let the familiar deliver to the crossbow instead.
But even with the bolt there, the crossbow is not reloaded. The interact action is to draw back the bowstring and load it to be fired again. Giving your familiar Manual Dexterity (PHB-218) to perform manipulate actions gives it the ability to Interact with the crossbow to load it.
Putting these together:
The familiar uses its first Interact action from Valet to fetch the bolt from you. Then its second Manipulate action to actually load the crossbow. This means that your action economy is still either [Attack, Reload, Attack] or [Attack, Command (valet + reload), Attack]. The next round will need to start with Reloading either way.
For Alchemist in particular, Valet may be a way to draw bombs or thrown weapons more efficiently, but it doesn't look like it helps with crossbows.