The short answer to this is no. From the DVSA’s perspective, the MoT is linked to the date of registration, so a 1953 Morris Oxford would not require an MoT - even if modified. However, if you modify the car extensively you will be required to re-register it. If you want to keep the original registration number your vehicle must meet a certain criteria. This is currently done on a points-based system and you'll need eight point or more, with five coming from having the original chassis, monocoque bodyshell or frame. Here's how the points are currently divided:

Chassis, monocoque bodyshell (body and chassis as one unit) or frame - original or new and unmodified (direct from manufacturer) 5
Suspension (front and back) - original 2
Axles (both) - original 2
Transmission - original 2
Steering assembly - original 2
Engine - original 1