"The DM might..."
These are not rules in the traditional sense. These are suggestions a DM might use to facilitate a subclass change in their game. Under "Training Time", we see (emphasis mine):
To change your subclass, your DM might require you to spend time devoted to the transition, [...]
The DM might also require an expenditure of money to pay for training, [...] This cost might be accompanied by a quest of some sort.
And under "Sudden Change", we see (emphasis mine):
Sometimes a character undergoes a dramatic transformation in their beliefs and abilities. When a character experiences a profound self-realization or faces an entity or a place of overwhelming power, beauty, or terror, the DM might allow an immediate subclass change.
These are not rules. These are suggestions to help a DM facilitate a subclass change, which is a pretty significant change. The idea here is to make the transition just as meaningful as where you came from and where you're going.
How would I personally work a subclass change?
I would ditch the limitation of "you can change your subclass when you would normally gain a new subclass feature."
The two methods offered give us distinct approaches for handling two different situations:
- A character says they want to switch subclasses some time in the future.
- A character says they want to switch subclasses very soon.
In either situation, I would work the subclass change at their next level up. In the first scenario, if they give me time to work in some plot, there is going to be some significant plot leading up to their next level up that will culminate in a change of subclass when they finally level up.
The second scenario is when I would use the "Sudden Change" guidance. If we are nearing a level up, and my player says "I want to change subclasses", I don't have time to work in significant plot build-up to the change, so I will opt for something like is suggested in this section at the next level up.
In either case, I am looking forward to the next level up, not the next subclass feature.
Waiting for a level up is how I would first express this to the player. If they felt that was too long to wait, I would of course see how we could work a transition even sooner.
What about rogues?
I present rogues as a use case for why I would break from the guidance given in the book. Rogue subclasses have a huge gap in features. Once a rogue gets their first subclass features at 3rd level, they don't get another until 9th level.
Say one of my players is a 5th level rogue and wants to change subclasses. I'm not waiting till 9th level to allow this. In this case, I would ignore the suggestion given in Tasha's Cauldron of Everything and still work the subclass change for their next level up.
Best Answer
It’s up to the DM, but the language does not contain an explicit restriction.
The Changing a Skill section is actually quite short:
That this is entirely up to the DM is hard coded into this rule. That said, the language does not contain any restriction on which skills may be replaced, it simply says “sometimes you pick a skill”.
So this isn’t like an ability score improvement where I just pick what I want. I don’t have to talk to my DM when making a decision about an ability score improvement. Working with your DM on this is just part of the process.
Personally, I see no reason whatsoever to not allow you to change any skill proficiency you want. But your DM may see a reason, and that is okay too.
It should be noted that the Basic Rules encourage Customizing a Background:
This further supports the idea that customizing your character is something that should be encouraged, including changing skills from your background.