This is relatively easy to achieve with the excellent ExifTool. The process should be similar for both Windows and MacOS (since the tool is available for both). Given that you have the executable installed and available on the path, you'd browse to the directory where the images in question are (assuming that there aren't other files there), open a terminal there and do the following:

exiftool "-DateTimeOriginal<filename" -d "%Y_%m_%d_%M.%%e" "-FileModifyDate<filename" -d "%Y_%m_%d_%M.%%e" .

The part with "-DateTimeOriginal<filename" -d "%Y_%m_%d_%m.%%e" tells it to read the current filename, parse the time from it and write it into the DateTimeOriginal tag; likewise with FileModifyDate, only that it writes to a different meta field. The . at the end means "the current directory" (so it would process all files in it), alternatively you could point it to another directory (with an absolute path or with a path relative to the current one), or to just a specific file with the -filename switch.

If your filenames differ, you can change the "%Y_%m_%d_%m.%%e" part accordingly. Of course this assumes that all the files there are named in a similar manner; if they're not, you would have to split them and process them separately, or rename them first so that they all follow the same pattern and then process them in one go.

Here are all the placeholders that you can use (copy-pasted from the site):

%a  - abbreviated locale weekday name
%A  - full locale weekday name
%b  - abbreviated locale month name
%B  - full locale month name
%c  - preferred locale date/time representation
%d  - day of month (01-31)
%f  - fractional seconds (see note 1 below)
%H  - hour on a 24-hour clock (00-23)
%I  - hour on a 12-hour clock (01-12)
%j  - day of year (001-366)
%m  - month number (01-12)
%M  - minute (00-59)
%p  - 'AM' or 'PM'
%s  - number of seconds since the Epoch, UTC (see note 2 below)
%S  - seconds (00-59)
%w  - weekday number (0-6)
%W  - week number of the year (00-53)
%x  - preferred locale date representation
%X  - preferred locale time representation
%y  - 2-digit year (00-99)
%Y  - 4-digit year (eg. 2006)
%z  - time zone in the form +/-hhmm (see note 2 below)
%Z  - system time zone name (see note 3 below)
%%  - a literal '%' character

If you have the hour besides the minutes, you can add it too.

By default exiftool creates a backup copy of each image it processes; you can disable that by adding -overwrite_original to the command.