What is the easiest way to install multiple 5A BS546 sockets next to one another?
Yes those sockets fit on normal single gang backboxes. 

The easiest solution is probably to just fit surface mount plastic boxes and then run normal twin and earth cable (any size from 1mm2 to 2.5mm2 will be fine) between them. No need for conduit. Surface mounting is easier both because you don't have to make a hole in the wall and because surface boxes are the full size of the faceplate so you can just butt them up (flush boxes are smaller than the faceplate so if you want the faceplates to sit next to each other you have to space the boxes just right).

Normal surface mount boxes have knockouts at the back of the sidewall (intended for use with cables that are directly clipped to the surface of the wall). Be careful when knocking them out, it's easy to break the boxes.

You can get backboxes designed to take two single gang accessories on one box. I've also seen boxes for one single and one double but i've never seen one for three singles.

Another option is "Euro modules". While I have never seen a double or triple 5A socket as a single unit you can get 5A sockets in Euro module form and install them in a Euro module faceplate. This would allow you to put three 5A bs546 sockets on one plate, but you will likely have to order the parts from specialist suppliers. 5A BS546 Euro modules and 3 gang (6 module) Euro faceplates are both fairly unusual items.

Unlike Americans we don't normally clamp cables where they enter and leave back boxes. We also don't have prescriptive rules for box-fill. We do however sleeve our earth conductors which Americans don't.