Featured in a "best of" anthology between around 2006 and 2015, not an old story.
The ending is a spoiler so:
For some major events which could have gone both ways it happens that the chronoscope shows both possible versions. The first night he sees the resurrection as portrayed in the bible and is jubilant. The second night he goes to view it again and is found by his boss drunk having seen the disciples deciding to make it all up.