Switch to face mode in the UV editor, or make sure that UV Sync Selection is disabled, then switch to island mode or set sticky selection mode to Disabled. All are options on buttons in the menu bar of a UV editor window.
The problem was that you used object coordinates. Object coordinates always point towards world X, Y and Z-axis. To fix this you have to either change it to UV or Generated coordinates which bend with the object.
SET TO GENERATED:
SET TO UV:
We, people from forensic department ... we already know :)
But yes ... please use more illustrative screenshot like - switch to edit mode to see wire (topology) of your mesh with probably more distance so we can get more context.
If I'm right ... just select top edge loop Alt+Shift
And Bevel Ctrl+B it (with Middle mouse add more loops.
Or use Bevel ...