Perhaps Exposure Program is not present in the file's EXIF metadata. Upload the photo to Dropbox or similar, post the sharing link here, and we can see what's going on.
Good point and excellent idea. Using ExifTool.exe I do see it has "Program Mode", instead...hmmmm, odd that Lightroom wouldn't have that option for display or even in the detailed list on the margin.
Maybe you can see something I haven't. https://www.dropbox.com/s/cv0idxs5axofb33/DSC_0124.NEF?dl=0
My version of ExifTool (10.46) has the exposure program marked as "not defined".
To build on ManiacJoe's reply, Exiftool shows three fields with the word "program":
The industry-standard EXIF:ExposureProgram field contains "Not Defined", so LR evidently shows nothing in that case. I have a several D5500 .nefs in my catalogs with EXIF:ExposureProgram set to other values and displayed by LR.
The industry standard only defines a fixed list of 8 different values for ExposureProgram. So perhaps if you choose a program in the camera that's not in that list, it sets ExposureProgram to Undefined.
The other two fields containing "program" are proprietary MakerNotes fields. LR doesn't show such fields, since they are undocumented by the camera manufacturers and specific to each make (and sometimes each model).
Hmmmm. I'm using the same version (10.46) and didn't even see that listed. I'm just using command line, though, and may have missed something.
Thanks to one and all for the responses and the help. Sadly it looks like I'm no closer to a solution to my issue but at least I know the cause...