7 talisman or spellhearts, 11 with feats.

The rules for talismans state that:

You must be wielding or wearing an item to activate a talisman attached to it. Once activated, a talisman burns out permanently, usually crumbling into a fine dust.
Each talisman’s stat block indicates the type of item it can be affixed to. Affixing or removing requires using the Affix a Talisman activity. A single talisman can be affixed to only one item at a time, and an item can have only one talisman affixed to it at a time.

Spellhearts behave the same way:

Spellhearts are permanent items that work similarly to talismans. You affix a spellheart using the Affix a Spellheart activity, which is otherwise identical to Affix a Talisman. The limit of one talisman per item remains—an item can have one spellheart or one talisman, not both.

I can have an indefinite amount of talismans attached to various equipment, but for the purpose of this question I'll only count talismans that I can activate at any time, that is, talismans affixed to items that I am "wielding or wearing".

The rules for carrying and using items have this say about wielding and wearing items:

A character carries items in three ways: held, worn, and stowed. Held items are in your hands; a character typically has two hands, allowing them to hold an item in each hand or a single two-handed item using both hands. Worn items are tucked into pockets, belt pouches, bandoliers, weapon sheaths, and so forth, and they can be retrieved and returned relatively quickly. Stowed items are in a backpack or a similar container, and they are more difficult to access.
[...]
Some abilities require you to wield an item, typically a weapon. You're wielding an item any time you're holding it in the number of hands needed to use it effectively. When wielding an item, you're not just carrying it around—you're ready to use it. Other abilities might require you to be wearing the item, to be holding it, or simply to have it.

Unreasonable Reading

Say I have a dagger that's sheathed on my belt, meaning it's being worn, by the loosest reading of talisman rules that's sufficient to activate a talisman attached to the dagger, because I'm "wielding or wearing" the dagger. Likewise, if I hang a padded armor off my belt I'm also wearing it, and that's sufficient to activate a talisman affixed to it. By this reading, I can have an indefinite amount talismans that I can activate at any time: I could wear a lot of light weapons (talismans affixed to weapons typically trigger off some part of a strike so they wouldn't be practical, but I could activate affixed spellhearts without wielding these weapons). However, it's self-evident that this reading must be wrong, and that this bit about ambiguous rules applies:

Sometimes a rule could be interpreted multiple ways. If one version is too good to be true, it probably is. If a rule seems to have wording with problematic repercussions or doesn’t work as intended, work with your group to find a good solution, rather than just playing with the rule as printed.

So when the rules for talismans says that I "must be wielding or wearing an item to activate a talisman attached to it" there's an implied "as appropriate for the item in question". In other words, I must wield weapons or shields and must properly wear armor (not just hang it from a belt or stuff it in a bandolier).

Items

There's a few odd items to address, before we count how many talismans I could activate at any time.

• Free-hand weapons such a gauntlets. The trait states that:

  When you're not wielding anything and not otherwise using the hand, you can use abilities that require you to have a hand free as well as those that require you to be wielding a weapon in that hand. Each of your hands can have only one free-hand weapon on it.

  So while I can affix a talisman to my gauntlet, I can't activate the talisman while also wielding something in that same hand.

• Attached weapons such a shield bosses and reinforced stocks. The trait states that:

  An attached weapon must be combined with another piece of gear to be used. The trait lists what type of item the weapon must be attached to. You must be wielding or wearing the item the weapon is attached to in order to attack with it. For example, shield spikes are attached to a shield, allowing you to attack with the spikes instead of a shield bash, but only if you're wielding the shield. An attached weapon is usually bolted onto or built into the item it's attached to, and typically an item can have only one weapon attached to it. An attached weapon can be affixed to an item with 10 minutes of work and a successful DC 10 Crafting check; this includes the time needed to remove the weapon from a previous item, if necessary. If an item is destroyed, its attached weapon can usually be salvaged.

  The item and attached weapon can each be affixed with a talisman, because there's no indication that the attached-weapon+item pair count as one item, if anything the bit about salvaging the attached weapon from a destroyed item suggests otherwise. Although, not stated explicitly, an attached weapons is also wielded (or else it would be impossible to strike with it), so if I'm wielding the item, I can activate the talisman affixed to it as well as the one affixed to the attached weapon.

• Both the handwraps of mighty blows and the bracers of armor allow affixing talismans, but both items have the invested trait, which states:

  A character can wear only 10 magical items that have the invested trait. None of the magical effects of the item apply if the character hasn’t invested it, nor can it be activated, though the character still gains any normal benefits from wearing the physical item (like a hat keeping rain off their head).

  Given that affixing talismans is not a normal benefit of handwraps or bracers, I must invest the handwraps of mighty blows and the bracers of armor for their ability to be affixed with talismans to work.
  Moreover, these items can be worn with any other item that does not have the usage "worn gloves/bracers", including armor.

So in total I can properly wear armor, properly wear and invest handwraps of mighty blows, properly wear and invest bracers of armor, and wield two weapons/shields each with an attached weapon, for a total of 7 items and therefore 7 talismans or spellhearts that I can activate at any time.

Feats

I have identified a few feats that can further increase the maximum.

• Talismanic sage and Grand Talisman Esoterica each allow for one additional talisman or spellheart.

• Juggle states:

  You can start to Juggle with a single stored item, an item you were carrying, or an unattended item within your reach. The item must be of light or negligible Bulk. You Juggle the item until the end of your next turn, unless you use this feat against next turn to continue Juggling (see below). While you are Juggling, you can use this action again to add another item to the Juggle, which must meet all the same requirements. You can Juggle a number of items equal to one more than the number of hands you are using to Juggle. [...] As long as you’re Juggling fewer than your maximum number of items, you have a free hand. You can wield weapons you are Juggling (but not shields or other items) as long as they can be wielded in one hand. For example, while you Juggle a dagger in one hand and wield a shield in the other, you are wielding the dagger but still have a free hand to Cast a Spell with a material component. Juggling two or more weapons with one hand doesn’t allow you to use feats that require two weapons each held in a different hand.

  So with this feat I could wield 2 light weapons with one hand. Unfortunately, the lightest combination of weapon and attached weapon totals 2L (a light firearm or a crossbow with either a bayonette or a reinforced stock), which is too heavy to juggle. That said, the Juggler archetype has two feats that further increase the maximum number of juggled items, so instead of wielding a weapon/shield with an attached weapon, I could juggle and wield 4 light weapons, thereby increasing the maximum number of talismans or spellhearts by 2.