Update: following comments I asked a friend who lives nearby to get up-to-date information. He took a picture of the actual Barnwell junction, which shows that @DavidRicherby and I are both right: the actual rail junction can no longer be used, but the rails and the points/switch and their actuation mechanism at the junction still exist.
He looked through the trees on the right to find out what is beyond the gate and saw a glimpse of rails, but it is private property so could not get any nearer. He also went around the other side to view from the meadow to the north, and there is no longer any track behind the factories, only some rail ballast. He spoke to a construction worker who confirmed the track was picked up. Where the construction work is happening (perhaps the Chisholm cycle path) my friend saw no evidence of any previous railway.
So if there is any track left at all, it will be a fragment on private property.
The station Barnwell Junction on the Wikipedia map you linked still exists (now a private residence), and the rail track at this part still exists, but not very far. This first section remained in use until recently to service factories and a fuel depot.
Parts of the line from there to Isleham and Mildenhall are marked on the 1:50000 Ordnance Survey maps as "dismantled railway" or "dismtd rly".
OS maps can be viewed online as streetmap.co.uk and a fragment of the route can be seen from a Mildenhall search, just below bottom left. Best viewed at 1:50000 scale.
Here is a clip from an OS map of the route from Barnwell Junction, which is at the bottom left of the image.
Here is the same route as seen on Google Earth.
The route crosses High Ditch Road under a bridge that still exists, then the A14 highway crosses it with no bridge, as the railway was already defunct at the time of its construction. It is likely that there are other parts which remain, and you can find them using a combination of these tools, possibly with Google Streetview too, to see where bridges still exist.
It is unlikely that any of the rest of the track itself still exists, and in places the route has been built over or ploughed. In other places is it used as farm tracks, and there are hints of this on the OS maps, guessed from their location and direction. From Google Earth, traces of the route can be seen even where ploughed over.
Below is a clip from Google Earth, showing the former Barnwell Junction station (at the bottom) and the railway track running past its left side towards the factories, where it stops. The tracks on the left are the main Cambridge to Ely route.