I've checked manuals I have on my computer for the Nikon D7200, D500 and D850 and they do not offer a date/time stamp option. The D3500 (and Powershots) do. A problem with adding a date stamp this way is that the stamp info is permanently added to the image - you can not remove it in the future if you want to print an image with the date.
As noted, the full date/time is embedded with the photo in the EXIF data. However, as far as I can tell, Lightroom export does not have an option to place the date/time on the. exported photo - it is just maintained in the EXIF data. The Lightroom print module does allow you to add the date to the white space around the photo. I could find no way to add the time nor a way to place the date/time over the actual image. Maybe there are options I am missing.
In the OP's original question, I am left with the impression that the person wanting the date/time info on the photo will not have a lot of experience with processing digital images. So, while FastStone Image Viewer will work (unless you use a Mac computer), it may not be a suitable solution for that situation.
I am surprised that the print shop didn't know date/time is in the EXIF of the photo. And, that their software didn't allow the option to stamp the data/time on prints.