The first step of disassembly is to remove the actuator lever and panel nut from the front. Then remove three screws from the front of the case and withdraw the front cover. This will expose the cam actuated switch contacts. There are three long screws passing through the case from front to back that electrically connect the contacts to wire terminal screw lugs on the back of the case, as well as holding the rest of the case together.

This is your opportunity to clean these contacts (see addendum below). For the MGA 1500 these contacts carry only the current necessary to operate the turn signal relay, less than one amp. For the MGA 1600 and MK-II, these contacts carry the current necessary to directly power two turn signal bulbs, about 3-1/2 amps. In either case, the contacts are adequate to carry the necessary current, such that they should last many years without burning or pitting, so a little dressing is all that will be required to clean them up for continued service.

Addendum, May 30, 2019:
For cleaning contacts, insert a bit of polishing cloth (with a touch of solvent perhaps) between the contacts, rotate the cam to bring the contacts together, and move the cloth to clean the contacts. This was previously noted as a piece of fine emery paper, but David Brown suggests: "My experience in the telephone industry in the 1970s tells me that these are high quality contacts and they are neither tarnished or pitted. They are certainly made of, or plated with, a rare metal, probably platinum. My recommendation is, if they are working, leave them alone. If they are not making contact then polishing with a clean duster or soft cloth should do the job. Emery paper will embed abrasive particles in the contacts – this will degrade a previously high spec component".