As stated, the drop rate per pack isn't known, and is just listed as under 1%.
@Cheese9Man You bring up a good question, and there was an update to the FAQ to address the thematic packs, which indicates that they're counted the same as regular Apex packs for the pack counter.
"How do the Bonus Heirloom Shards work?
When a player opens a Thematic Pack, there is a <1% probability of Heirloom Shards dropping. Heirloom Shard drops do not deduct a Thematic Pack from your inventory. Heirloom Sets cannot be crafted with Crafting Metals and can only be obtained by exchanging Heirloom Shards (or through a Collection Event). A player cannot open more than 500 Thematic Packs or Apex Packs without receiving Heirloom Shards (assuming the player does not own all of the available Heirloom Sets). Once a player owns all of the Heirloom Sets, the player will not be eligible for this bonus until more Heirloom Sets are added to the game."