How to find location on Cambridge-Mildenhall railway that still has tracks/rails?

For the UK, Ireland, and the west coast of the US, I know no better online resource for this than Rail Map Online. It allows you to view all the railways that have ever existed in the UK overlayed onto various maps (including Google Maps and OpenStreetMap).

I don't know about the specific railway you mention, but there were a lot of railway closures in the 1960s and I don't know of any where the rails still exist. Quite often they have been converted to footpaths/cycleways.

My understanding is that the rails were generally sold for scrap rather than abandoned.

EDIT Looking at Mildenhall station and the old route confirms that (at least most of) the rails are no longer there. https://www.openstreetmap.org/#map=17/52.33656/0.50065&layers=HN

The National Library of Scotland has digitized a lot of old UK maps (and some world-wide). It will display the old map overlaid with a new one, with a range of dates for the old map. The new map can either be a satellite, road, or OS.

The range of old maps is better in Scotland, but elsewhere in the UK there is a reasonable selection available.

The site makes it easy to visualise the old map in relation to current maps. If you use an old map between about 1920 and 1963 (the beeching cuts), it will show the old railways. Later maps will show the route of old railways until they have been built over.

For example, https://maps.nls.uk/geo/explore/#zoom=14&lat=52.2190&lon=0.2069&layers=11&b=3 Change the transparency (bottom left) from 100% to 75%

It is also currently shown in Open Street Map.
https://www.openstreetmap.org/relation/317247 But, like most if not all the others, it does not indicate whether the actual steel has been removed. I looked at a lot of aerials, and in the few places possible, Google’s street view and found no rails. But three photos in someone’s comment showed some.

If you are after locations where the track itself is still present, then you're out of luck - Disused Stations suggests that the track was all lifted on closure except for that accessing the oil depot mentioned in @WeatherVane's answer.

This is generally the case for most British disused railways - the track was generally lifted fairly quickly after closure, unlike other countries where lines were often mothballed.

If you just want to find the route, then OS maps are generally the way to go, as they generally show disused railways, as shown in the example above - they also show rights of way so that you can find how to access the line. Quite a lot have been converted to long-distance foot or cycle paths and so are easily accessible.