The car will not drive?

The description sounds like a combination of low fluid and internal wear. The low fluid can move around, and the filter doesn't draw in enough which aerates the fluid and with the air in the system it creates more problems. Then the vehicle has to set a while for the air to come out of the fluid. The first thing to check would be the fluid level in the transmission, if it hasn't been serviced in a while you might consider a full fluid and filter change. Another possibility is a combination of low fluid and a failed seal on the filter, so it draws in air above the fluid when it sloshes during those maneuvers. The solution would be the same, service the transmission.

This guide might help you with that:
https://www.2carpros.com/articles/how-to-service-an-automatic-transmission

To check the fluid level drive it some, now stop in a level spot and with your foot on the brake shift through the gears. Now with the transmission in park pull the transmission dipstick. The fluid level should be in the hot area. It should also be relatively red in color, brown or smalling burnt is a sign it needs service. I would plan on changing the fluid and filter and then see if the issue is solved.
You need filter 3533012040 and about 3 liters of Toyota ATF - T-IV
plus, the pan gasket 3516812060.
The basics are in the guide above. You use the drain plug to drain out the fluid. Then remove all but 2 bolts with a drain pan under the transmission. Now remove the last two bolts and let the pan down so the rest of the fluid can drain. Now remove the filter and gasket and clean the spot the filter attaches. Replace with the new filter, wipe out the pan really well and remove the old pan gasket. Check the bolt holes in the pan to be sure the lip is flat. Install the gasket and pan and tighten the bolts to 48-inch pounds. They don't show a sequence, but a simple cross pattern should work. Now refill the transmission and check it cold with the dip stick. Now run the transmission through the gears and drive it to heat up the fluid. Check the level and add as needed. See if the problem is gone.

Okay, you have the MMT unit. Basically a 5 speed manual with a robot clutch. On those the actuator that moves the clutch can get hot and it locks the transmission effectively into neutral. Then it has to cool down before it will operate again. Usually, it will flash the N light and set a trouble code (P0810) in the TCM. They didn't offer that transmission in the US in that vehicle for that year because we only got the Corolla while the Verso is actually a different platform. As such I know what the issue is courtesy of some friends in Europe, but I have no service information on that system in that particular car.

The folks I deal with are diagnostic techs, they go visit a shop, diagnose the issues and then leave, you would be better served to find a shop local to you who you can trust and have them work on it. The issue then is that like here in the US, finding a shop you can trust is the difficult part. I can say that the folks I know really dislike that transmission, and the version that did make it to the US has the same dislike here as well.

The only real solution is to replace the actuators that power the clutch assembly. They are starting to fail, which is what is causing the issue.

Wish it was better news but it's unfortunately a known problem. Similar to the CVT transmissions when they used them in US vehicles. They were fine in lighter cars with smaller engines and a generally "lighter" population. Then they take the same transmission, put it with an engine that has twice the power of the original, add on all the extra safety and emissions items for the US and then put 3 large Americans in there and then wonder "Why are they failing to carry twice the load they were rated for?"