https://www.screwflix..com/p/masterplug-13a-4-gang-switched-surge-protected-extension-lead-1m/1601g
Would that be acceptable to have the washer and tumbler plugged into if I only run 1 at a time?
You dont need a new extension lead. You need more sockets. Your problem is that your double socket is rated at 13A. I believe you said that the fuse in your orignal extension lead blew. That would have also been 13A. So basically you are still overloading your socket if you run all the appliances.