Then you must have done your survey with an older version of ReachView. It has been a feature in the stable version for a month now:
Are you sure that GeoJSON doesn’t contain those start and stop times? If not, then you’ll just have to do it visually with RTKPLOT. Click the bullseye icon to let you select a specific point in your plot. Click over one of your survey points. Then switch views from ground track to position where you see the three graphs (lat, long, height). There it is easy to see where you were stationary on a point because it is a timeline view. Just slide the selected point to the start of a point and write down the time. Then slide it to the end of a point and write down the time. Then limit the view to with the start and stop times. Then write down the average coordinate from the statistics info in the top right corner of RTKPLOT. Do that for each point.
If you haven’t upgraded ReachView, then do so. That way you will get the start and stop times in your survey project CSV file next time and doing this will be a lot easier.