Edge 1040 does not show lap summary when pressing the lap button as appeared on my previous edge 520+

There is something wrong with the profile you are using as the lap summery should be working.

Is the profile you are using one that was imported as part of the initial set-up process?

Yes, most of the edge 1040 profille were  transferred or added automatically from my existing connect profile.

Do you have a recommendation how to solve it?

The easiest from a technical point of view would be to delete the profile and manually create a new one on the unit.

Do you mean creating a complete new profile (new user etc.) which means that all my existing data would not be transferred or just delete it from my edge1040 and create it again with the same user etc.?

If you mean a complete new user, do you have any other option while I am keeping all my data?

What I mean is a  new activity profile only.

Goto Activity Profiles -> Create New

Then try taking a lap split using that new profile and see if the lap banner comes up.

If you don't have many profiles I could try and manually fix them by editing the profile contents, but that is not something you could do.