Potion Miscibility: does the DM choose the permanent effect

HP are an Abstraction

First, remember that HP are not just a pure measure of how much physical injury a creature can take before being reduced to chunky salsa. It is an abstraction of the creature's ability to avoid death. If a Fighter hits a Goblin for 20% of its total HP, that doesn't mean the Goblin's left leg was cut off or disabled. It means the Goblin twisted out of the way, taking a light cut to its arm and tweaking its leg. It is hurt and distracted, but not sufficiently injured to impede its ability to fight.

Similarly, the 6d10 force damage for a botched potion mix does not necessarily equate sitting on a brick of C4 as it goes off. Even though the damage is typed 'force' for the purposes of resistances and weakness, some of the actual 'damage' might be the headache, fuzzy vision, and ringing ears from the explosion. So the reason taking the explosion inside still does the same damage is because the abstraction shifts. Even though the PC is hurt more by the explosion inside, they don't see or hear the explosion so no fuzzy vision or ringing ears.

Does it even Work?

Arguably, mixing two potions in your stomach would not even cause this. It is not just the mixing of the ingredients that causes this, but the magic they contain. When you drink a potion, the magic is immediately imparted on you. You do not have to digest the potion first. So the magic is gone and you are left with a non-poisonous beverage. So by the time you drink the second potion, the two magics aren't mixing and there is no more harm than having any two or three or seven buff spells cast on you at the same time.

You might counter, what if the person drinks both potions at the same time? RAW says they can't. Even if a class ability or feat allows them to drink two potions in a single action, presumably they are still being chugged quickly but separately. In fact, if the class ability/feat does not specifically say this is an option, that would tend to be a strong indicator it is not.

What's good for the goose...

If you decide to rule that drinking two potions together results in a mixing scenario, and that the 6d10 damage result is instant death for the imbiber, be ready for the players to use it against you. Watch as they pour dozens of potions down a sleeping dragon's throat until one pair mixes badly, instantly killing it. Marvel as they slip potions into a wine tasting event to gorily kill the king and his court.

A Rogue cannot use Fast Hands on any magical potion

Magic items do not require the Use an Object action to be activated, but a separate action entirely

The following quote directly addresses how this feature is limited:

[...] If an item requires an action to activate, that action isn't a function of the Use an Object action, so a feature such as the rogue's Fast Hands can't be used to activate the item [...]

- Dungeon Master's Guide (page 141: Chapter 7 - Treasure: Magic Items: Activating an Item)

Mundane items require the Use an Object action to be used

Do note that the above quote applies to magic items and the general quote which applies to mundane items is found in the description of the Use An Object action:

[...] When an object requires your action for its use, you take the Use an Object action [...]

- Player's Handbook (page 193: Chapter 9 - Combat: Actions In Combat: Use An Object)

One specific example given, that is similar to drinking a potion, is that one can drink all the ale in a flagon using their object interaction.

Potions are magic and explicitly need to be "activated"

The question then is whether a potion requires an action to "activate", luckily the nearby sections on "Activating a Magic Item" and "Consumables" state the following:

Activating some magic items requires a user to do something in particular, such as holding the item and uttering a command word, reading the item if it is a scroll, or drinking it if it is a potion [...]

- Dungeon Master's Guide (page 141)

Some items are used up when they are activated. A potion or elixir must be swallowed, or an oil applied to the body. The writing vanishes from a scroll when it is read. Once used, a consumable item loses its magic and no longer functions [...]

- Dungeon Master's Guide (page 141)

Both of these quotes establish that drinking/consuming a potion counts as activating it, and we know that a potion of healing requires an action to be consumed and that it is a magic item, and so its activation is not a function of the Use An Object action. Thus, the Rogue's Fast Hands cannot be used to activate (drink/consume) a magic potion.

Though drinking a flagon of ale and drinking a potion seem incredibly similar there is a key difference: one is magical. How this difference looks like narratively, or why magical things are harder to drink is up to your GM.

That said, Fast Hands can still be used to draw the potion

We already have the following question and the highest-scoring answer there says:

• Can a character grab a potion from a pack and drink it without using their action?

[...] Thus you don't have 2 instances of interact with object going on, you have 1 interact with object (drawing the potion), and a second action that is specific to the magic item in question. [...]

What this means for our Thief is that they could replace the object interaction used to draw the potion with their bonus action from Fast Hands. This would let them still have an available free object interaction to use, likely during their movement.

Of course, they could also have replaced the object interaction during their movement with the bonus action from Fast Hands and used their free object interaction to draw the potion. These are, as far as I'm aware, effectively the same.