Is it obligatory for someone afflicted with urinary incontinence to wait for it to stop in the bathroom, or can they stand up? And what are the implications of that regarding washing what it has stained?

We ask Allah Almighty to grant you healing. You are not required to remain in the restroom if you can protect your clothes or body from being soiled by urine. If standing leads to contamination with impurity unnecessarily, then it is not permissible. Any urine that touches the body or clothing must be washed until the impurity is completely removed.