I have used the Shop-It! app, from Just-shop-it.com, that I found in the play store probably 10 years ago. I have transfered it from one phone to another for years, but they no longer exist, the website it gone, it's no longer in the play store. When I have to change phones again I can only hope that it transfers again without a problem.
I have looked several times for other apps for helping with mystery shopping, but they seem to have all gone by the wayside.
I have forgotten to mark time and had to go searching for a time on my phone. text messages have the time of each text, from each of the parties, I text and/or PM on messenger to myself, when I need to remember something. And if I neglect to mark my end time or even the start time of a shop (start time isn't usually a problem), I can look at my Google Maps timeline to see when I arrived and departed from a given location.
I have a Google Pixel, so the clock stopwatch function displays days:hours:minutes:seconds, but even by taking a screen shot, the end times come out blank. I've learned that with the stopwatch you have to hit "lap" at the end, or the end time that you need will be flashing and the screen shot won't capture it. And then you can do a screen shot to keep your times and all of them will show up.