After testing this scenario out, I can confirm that it is possible to change the format of a "Work Date" field type to include the time. I can also confirm that the predecessor fields do not appear to be affected by the field type change.
You will want to edit the field properties of the "Work Date" field in question and change the field type to "Date/Time".
At this point you will have to edit your existing records to adjust the times you would like to associate. Using the "Grid Edit" view you might be able to update your records faster than editing each record individually. New records generated will display the appropriate time.