The freeware program ReNamer is used to simultaneously relabel all camera trap pictures in folder.  ReNamer is downloaded from:
http://www.snapfiles.com/downloads/denrenamer/dldenrenamer.html

(1) Open Renamer.
(2) Go to Setting, General, then Meta Tags.  Change the date format to:yyyy mm dd hh nn ss without punctuation.  Make certain there is only one blank between fields.
(3) Click Save. This format is now permanently saved.

Two Rules are needed to relabel all camera trap pictures with their date and time.  The first Rule deletes the name of the jpg file; the second rule lables teach file with its creation date and time. These Rules must be created and saved.

(4) In ReNamer click + Add.
(5) Select Delete, in the Until box click Till the end.  Click on the + Add Rule box.
This Rule deletes the name of each jpg but since there can be only one file with a blank name ReNamer shows an error that multiple files will have duplicate names.  Ignore this error.

(6) Next click + Add, then Insert, then the yellow lightening bolt Insert Meta Tag.  Slide down to Exif_Date.  Then click on + Add Rule.
This Rule makes the label the creation Date and Time.

(7) On the main page select Setting, then Presets.  Give your rules a name and save them so you can use them every time. I suggest you name them according to the brand name of the camera traps. 

(8) To add a single file click on Add Files. To add a folder click on Add Folders.  All camera trap pictures from a flash card can be relabeled simultaneously.
(9) Clicking on Preview gives a preview of the new label.  Clicking on the green ReName button renames the file using the rules you created.  

If the date and time match that of the picture, then your rules worked.  If not, then the EXIF_Date did not work and it's another Meta Tag you need.  You might try creating another set of rules replacing EXIF_Date with the File_Date_Modified Meta Tag.