Disclaimer: I am not a lawyer, my interpretation of a partial reading of the he Road Vehicles Lighting Regulations 1989 could be wrong and must not be relied on!
Regulation 23, below, appears to be the one that basically say that if a lamp or device is there, it has to work properly.
However - there is a list of lights etc. to which this applies, and the ‘Reversing lamp’, which is defined in the Regulations, is not on it. So my interpretation is that it doesn’t need to work.
In practical terms, a reversing light is pretty handy to let people know that you are reversing, or want to reverse. Illuminating the way behind is less of a problem, just use the hazard lights.
I wouldn’t be taking the gearbox out. I might think about rigging a switch to use it manually, although that might not be as simple as it once was.
The link to the regulations is here
There are references to “Reversing lamp” in other places if you want to search for them. I just remembered why I didn’t want to be a lawyer.
Maintenance of lamps, reflectors, rear markings and devices
23.—(1) No person shall use, or cause or permit to be used, on a road a vehicle unless every lamp, reflector, rear marking and device to which this paragraph applies is in good working order and, in the case of a lamp, clean.
(2) Save as provided in paragraph (3), paragraph (1) applies to–
(a)every–
(i)front position lamp,
(ii)rear position lamp,
(iii)headlamp,
(iv)rear registration plate lamp,
(v)side marker lamp,
(vi)end-outline marker lamp,
(vii)rear fog lamp,
(viii)retro reflector, and
(ix)rear marking of a type specified in Part I of Section B of Schedule 19,with which the vehicle is required by these Regulations to be fitted; and
(b)every–
(i)stop lamp,
(ii)direction indicator,
(iii)running lamp,
(iv)dim-dip device,
(v)headlamp levelling device, and
(vi)hazard warning signal device,with which it is fitted.
(3) Paragraph (2) does not apply to–
(a)a rear fog lamp on a vehicle which is part of a combination of vehicles any part of which is not required by these Regulations to be fitted with a rear fog lamp;
(b)a rear fog lamp on a motor vehicle drawing a trailer;
©a defective lamp, reflector, dim-dip device or headlamp levelling device on a vehicle in use on a road between sunrise and sunset, if any such lamp, reflector or device became defective during the journey which is in progress or if arrangements have been made to remedy the defect with all reasonable expedition; or
(d)a lamp, reflector, dim-dip device, headlamp levelling device or rear marking on a combat vehicle in use on a road between sunrise and sunset.