I have a 15 Amp wall socket and recently bought a washing machine with a 13 Amp plug. Can I plug the washing machine into the 15 Amp wall socket using an 3 round-pin to 3 square pin adapter?
Yes, as long as the convertor in the middle is marked to handle at least 13 Amps then you will be fine
You can also cut off the original plug "South Africa?" and install a UK style style (at least 13 Amp) plug onto it without any problems.
Round prong wiring. Rare to contain a fuse unless you specifically buy one.
Although if you are using something that is going to draw near 13Amps (2500Watts+) then change the plug to the one you need. Using converters for heaters, stoves, hair dryers or other heavy appliances is a very bad idea. Do not even consider using it temporarily.