If some total dick has cut the reverse light wire/s then it will work just re joining them, as you have a towbar why not fix the wiring to it, as towbars are very useful to have, and wiring easy to do.
I would find the two pin connector on the reverse light switch on the gearbox, this will tell you the colours, and you can also probe for a live at the terminal(ignition on), if you have a live there bridge the two terminals and see if the lights work, if they do it's switch, or adjustment of switch that is at fault.
There is no connection to a trailer socket from the reversing lights, certainly NOT on a std black 12N socket, one can be put on if it has a extra GREY socket as can rear fog, soon to be part of mot!!
tabetha