Today, going south from the lake area, I paid special attention all the way, but did not find any section with artificial tolls, worrying that I might have missed it and not paid in advance ?

There is no toll road to the south of the lake district.

There really is a toll station, so it's unlikely to miss it.

It doesn't matter. All toll booths are manned, so just pay the fee if there is one, otherwise it's best.

I asked the car dealership, only the East of England has toll roads, Scotland and Wales don't have toll roads.

You don't need to pay in advance for expressway tolls. If you didn't pass by, it means there is no charge.

Then there is no charge, don't worry