In practical terms, 32
Because this is not for a theoretical optimization puzzle, but for your actual player character, I'll answer this for your specific character. First, what is a number of minions you could practically amass?
From Battle Smith Artificer 3: 2 constructs
Infuse an Item (level 2): will give you a maximum of two Infused Items. As you can have each of your infusions in one object only, the Humunculus Servant Infusion can be used with it to create 1 construct (cost 100 gp)
Steel Defender (level 3): allows you one defender at any one time, 1 construct
Replicate Magic Item: this does not do anything for you. (The Pot of Awakening creates plants, not constructs), 0 constructs
Artificer spells: none of the spells on your list up to spell level 2 creates a construct. The earliest is tiny servant at spell level 3, for which you would need to be an artificer 5. 0 constructs.
From College of Creation Bard 13: 30 constructs
Animating Performance (level 6): you can animate one Dancing Item, and at any time you can only have one of them active. It lasts for 1 hour. 1 construct
Bard spell list: this will give you access to animate objects (level 10, spell level 5). The spell requires concentration, and for up to 1 minute allows you to animate up to 10 tiny objects that count as constructs. It also is a good damage dealing spell. 10 constructs
Magical Secrets (level 10) lets you choose two spells from any list, at 10th level up to spell level 5. This opens up a huge space of spells.
You could for example (if the DM allows non-core spells, which is likely as Artificer is a non-core class) pick tiny servant. This is a third level spell with a duration of 8 hours that creates a construct, and you are not limited to how many you create, other than by your spell slots. As a bard of 13th level, you have 10 slots of 3rd level or higher, so at most you could create 10 of them per long rest. If you still have all of these slots at the end of the day and cast with unused spell slots at the end of a long rest, then cast them again on the new day, you can double up on that for 20. If you save one 5th level slot for Animate Object, for 19. 19 constructs.
Another option is summon construct but it only summons one construct for up to one hour, and is concentration, so it competes with your use of animate objects. The create homunculus (spell level 12, caster level 6) is not an option unless your DM allows you to delay the pick, because rules as written , you cannot delay to pick, and at 10th level only have access to level 5 spells. 1 construct (but not while also using animate objects)
This would bring you to a grand total of 32 constructs at peak, 2 of which would hang around all the time (1 homunulus servant, 1 steel defender).
Some considerations for your build and for minion management
Please consider that while constructs may be charming and fun intially, if you try to optimize your character solely for lots of them, that is unlikely to result in a well-rounded character. You will have to make trade-offs for this. For example, the Magical Secrets feature is very useful, and in my list, you burn both of your picks on just having more weakish minions, which adds little to your character. You already have humunculus minions, for example, so how much do you really benefit for having additional tiny servants? Would it make you play any differently? If you were not using it on even more constructs, you could pick other useful effects.
As @MivaScott points out, having more such construct minions also has diminishing returns. Many of these constructs require the owner's bonus action to actually do something; very few act on their own. Did you use your bonus action to create magic stones for the tiny servants? Too bad, now those tiny servants, are just going to stand/float around for the round. The steel defender is going to sit idle, like your Homunculus Servant. They may look like an impressive force, but they're just hit point sponges then.
Lots of minions also are not neccesarily a fun play experience, because they are a hassle to manage. I have been playing a wizard that is able to create a whole zoo of minions (with animate dead, a familiar, a Shield Guardian, and a caggle of elementals from summon minor elementals at times, and he also owns a pot of awakening and an ebony fly). I stopped using most of them, as it became too much of an administrative burden. For example, if you play on a virtual desktop, you have to move around all these pogs all the time, and at higher levels many of them contribute little to combat. See here for helpful advise on how to manage your minions.
