I hope this is helpful. I am posting this after a request, and despite the very flippant tone, I am not trying to take the pi$$...
There are four ways I have identified that one can wash using non-mixer taps -
1) Fill the sink using a mix of hot and cold water, and use that water to wet the soap then rinse
2) Use a flannel, having soaked it in a mix of hot and cold water
3) Turn both taps on and mix the water in your hands, taking care to use cold water first to avoid scalding
4) Just use the hot tap. Most of the time, unless you have exceptionally good heating, the water will be cold when it first comes out of the hot tap and will gradually get warmer. In this case, you can effectively pretend that the hot water tap is a mixer tap, and all will be well with the world!
Please feel free to add or amend.
Vicky