I want my potatoes to be baked within an hour, but I forgot to preheat the oven. Should I put them in while preheating or not?

In the UK, we call these 'jacket potatoes'¹.

As already mentioned, putting them in a cold oven as it heats might win you five minutes, but not much more. It's very slightly more efficient overall than waiting for the oven to be hot then opening the door.

Two additional methods of speeding up jacket potatoes.

1. Put a metal skewer through the longest centre, then bake in the oven.

2. Alternatively: Split the task, half microwave, half oven. Prick the skins with a fork first, or they may explode in the microwave. 15 mins* at full power, then 15 mins in the oven is a 'fair guess' at how long it will take. This, of course will vary depending on size/weight of potatoes, power of microwave & temperature of oven.
   *I think the last time I ever actually did this, I had an 850W microwave. Reduce time accordingly if you have a more powerful one.
   Though the mic should manage to have heated the centres in around 15 mins, you can still use the skewer trick once they're ready to go in the oven. The skewers would still be useful to test the centres are cooked in the mic before you transfer.

The essence of the microwave method is to get the centres cooked before you transfer to the oven, to add back some flavour.

I'd weigh up which method to try based on how much time you have. Skewers will bring the time down by maybe a third, so that might be borderline for your hour from cold. Mic + oven & you're down to half an hour or so, but the result isn't quite as good.

I suppose you could still use the microwave as a starter, even if your result is to be 'roast potatoes' ie, peeled, chopped, done in hot fat. It will be a lot messier, but should still be feasible. I've never tested this, so timings & quality of result you'd have to find out for yourself;)

^[1]Addendum
This may be cultural/linguistic, but to clarify from a UK perspective, these are baked (or jacket) potatoes… Comments suggest the same descriptions are true for the US.

…and these are roast potatoes

You can put the potatoes in the oven while it's heating, and that's not going to make any overall difference in how long it takes the oven to get hot. Opening the door to put the potatoes inside causes more heat loss than the mass of the potatoes possibly could. And then you still have to heat up those potatoes.

However, based on some personal experiments, it's also not going to save you much time.

If I put large russet potatoes into a 400F oven, it takes them 55-65 minutes to be fully baked. It also takes my oven around 20 minutes to heat up to 400F. If I put the potatoes in a cold oven, it takes them 70-80 minutes to cook ... meaning it only saves me around 5 minutes to skip preheating.

Obviously it still makes energy sense to skip preheating with baked potatoes. But it won't get dinner on the table in an hour.

Here's a hack: You can microwave your potato before baking it (like suggested in this great answer).

From The Spruce Eats:

The only problem is they take an hour to cook, which requires planning ahead. If you're trying to get dinner on the table in a hurry, there is actually a method you can employ: Use the microwave to speed up the cooking process, cutting the baking time in half. While microwaving to completely cook the potato will result in a soft and mealy Russet, cooking in the microwave for just 5 to 6 minutes before placing in the oven keeps the potato's original texture. Drizzling with olive oil and sprinkling with salt before baking at 400 F will allow the skin to get nice and crisp.

From Good Life Eats:

I'm not a big fan of only microwaving baked potatoes, because you don't get a nice crisp skin on them (instead they're a bit soggy). So, I microwave the potatoes before I bake them, and then finish up my baked potato in oven. This method is also great when you need to free up the oven for something else.

Two reasons for preheating an oven are:

1. Makes cooking times more predictable because different ovens heat up at different rates; all other things being equal, a given dish will take the same amount of time to cook regardless of what oven it's placed in, assuming the oven is already holding at the specified temperature (bearing in mind that ovens do vary in temperature calibration).
2. You want get the surface of the food browned/seared (if the oven is already hot, and hot enough).

An oven will heat more slowly with cold food already in it than if it is empty, but it's a small difference. On the other hand, the food will start cooking as soon as the oven is hot enough, so overall cooking time should be shorter (from the time the oven is turned on until the food is cooked); the difference can be hard to predict and would only be determined by trial and error.

I had the same problem and cut the potatoes in half making them as thin as possible. The heat has less distance to travel so the potatoes will cook faster. I spread a cookie sheet with olive oil, added salt and coarse ground pepper, and put the potatoes on the sheet cut side down. The large potatoes I buy as loose potatoes are done in 45 minutes at 425F with the cut side nicely browned. It is the only way I bake potatoes these days. It also makes it easy for people to take half a potato when a whole one is too much. My oven takes about 15 minutes to heat to 425F. I put the potatoes in for the last 5 minutes. The broiler element is used for the preheat, but it doesn't brown the skin too much in that time.

I would caution that it very much depends on your oven and how it pre-heats. My oven heats up by turning the bottom heating elements on at full blast, so if I put something in the oven while it's doing that it essentially broils it from the bottom. I typically bake potatoes directly on the oven rack, so that would definitely burn the bottom of the potatoes unless you put some sort of barrier between them and the bottom of the oven such as a baking pan.