I know it is always on the near-side, but is there any reason why it can't go on the off-side?
Nope none at all. I looked up the regs last year as someone was asking about the front plate. Here is the official answer for the rear, hope it helps
ROAD TRAFFIC
The Road Vehicles (Display of Registration Marks)
Regulations 2001.
Where it states relevant date it means 1st September 2001.
Fixing of rear registration plates: vehicles registered on or after the relevant date
5.(1) This regulation applies to vehicles, other than works trucks, road rollers and
agricultural machines, first registered on or after the relevant date.
(2) A registration plate must be fixed on the rear of
(a) the vehicle, or
(b) where the vehicle is towing a trailer, the trailer, or
(c) where the vehicle is towing more than one trailer, the rearmost trailer.
(3) Where a vehicle (or, in a case where the plate is required to be fixed on a trailer, that
trailer) has been constructed so as to satisfy the requirements of the relevant type-approval
directive, whether or not it is required by law to satisfy them, the plate may be fixed in the space
provided in accordance with those requirements but if it is not so fixed it must be fixed in the
manner required by paragraph (5).
(4) Except as provided in paragraph (3) the plate must be fixed in the manner required by
paragraph (5).
(5) This paragraph requires the plate to be fixed
(a) vertically or, where that is not reasonably practicable, in a position as close to the
vertical as is reasonably practicable, and
(b) in such a position that in normal daylight the characters of the registration mark are
easily distinguishable from every part of a relevant area having the diagonal length
specified in paragraph (6).
(6) The diagonal length of the relevant area is
(a) in the case of a mark having characters the width of which is at least 57 millimetres,
22 metres,
(b) in the case of a mark having characters the width of which is 50 millimetres, 21.5
metres,
(c) in the case of a mark having characters the width of which is 44 millimetres, 18 metres.