The best bet for oldest in situ rail is probably around the network in places that don't get touched too often.
As a general rule, main line rails get replaced every couple of decades less, with some curves and high stress locations being changed more often. This scales with speed, so modern high speed lines get swapped more often, but stuff along 125mph main lines gets changed less frequently. Also things like viaducts and tunnels, or over level crossings where road grit corrodes the rails, so some places get changed much more frequently.
On less used branch lines, so long as there aren't any heavy freight trains, the track can sit there for decades. Think the Looe branch, or the island line. Places like this have old track because there are low speeds, light trains and generally they're not used very much. There has been an effort to change to continuous welded rail across the network, but this is a slow process, and doesn't work on tight curves or in very remote areas (West Highland line for example).
Then there are the vestiges of old track around the network in yards ect. Places like Tyne Yard, Carlisle Yard and sidings outside stations (Carlisle and Dundee spring to mind) were layed in the sixties in some cases, and the track obviously hasn't been changed since. In these places, it might be that they are used so infrequently that it doesn't matter, or a rake of wagons has sat on them for decades and so they haven't been changed. There are loads of places around the network where the old layouts just aren't needed any more, so no work is ever done to spruce them up. Also rail lasts for a long long time, so in places with low traffic, it's not really urgent.