Restore Life takes 1 action plus 1 hour
The Restore Life option isn't the only one that takes longer than one action. There is also Major Transformation (emphasis added):
You can transmute one nonmagical object—no larger than a 5-foot cube—into another nonmagical object of similar size and mass and of equal or lesser value. You must spend 10 minutes handling the object to transform it.
Given that this use of the Master Transmuter takes longer than 1 action, there is no reason that other uses cannot also take longer.
Using the transmuter's stone is a 2-step process
Another way to establish this is to look carefully at how the transmuter's stone is used (emphasis added):
Starting at 14th level, you can use your action to consume the reserve of transmutation magic stored within your transmuter’s stone in a single burst. When you do so, choose one of the following effects. Your transmuter’s stone is destroyed and can’t be remade until you finish a long rest.
So the process is:
- Use your action to consume the transmuter's stone.
- Choose an effect and do whatever that effect says.
Some of the effects just happen automatically (Panacea and Restore Youth), while other effects require some action (Major Transformation and Restore Life). Among other things, this means that the stone is consumed immediately, and is not recovered if the effect is cancelled (e.g. if the casting of Raise Dead is interrupted). Consuming the stone's magic grants you the ability to cast the spell, but it doesn't cast the spell for you. You still need to cast it as normal.
(Also, you'd better have a 500 gp diamond on hand, because if I'm not mistaken, the ability doesn't negate the need for spell components.)
(Extremely technical note that will never come up in actual play: a multiclass wizard/fighter with action surge could start casting Raise Dead on the same turn they use their action to consume the stone, making the total time 1 hour rather than 1 round + 1 hour.)