There appears to be a currently (2017) extant route of A5125 in Loughborough, Leicestershire; though the evidence is confused and sketchy.
At the time of writing, the number does not appear to exist in the records of Leicestershire County Council. However, it appears on Department for Transport traffic count data, and also on Ordnance Survey mapping within their Streetmap product.
However, the OS mapping data is confusing at best. It shows the road starting on the A512 at the junction of Broad Street and Ashby Road, and follow Ashby Road in towards the town centre. where it then follows Ashby Square before terminating on the unclassified Swan Street. Whilst that looks partially reasonable, the facts on the ground add a layer of confusion. Broad Street, the continuation of the A512 is (despite the name) a relatively narrow one-way street, so taken at face value there is no direct access from the A6 to the A512 heading westbound. In addition, Ashby Square is also a one-way street forming part of a gyratory system, and the remainder is shown as unclassified. The road number is not shown on either Google Maps or OpenStreetMap at the time of writing, both claiming that this road is simply part of the A512.
Not helping the confusion, there is very little signage in the area to assist.