So I bought something off Amazon called "Luigi's Plunger". Filled the sink with hot water and gave it a go. It worked *too* well: decades of mould, dirt and hair filled the sink and flew all over me and the bathroom floor. Then I had to ladle the muck out of the sink...
Haha oops, I was using it the wrong way round. I should have expanded the plunger so it fills with clean water then *push* that down the drain, rather than sucking out of the drain. Now I did that, the slow flow problem is fixed