Thank you everyone who recommended Tulip and Bohin needles. I tried some, and I like both – on balance I prefer the Bohin, just because they’re more like what I’m used to. I like the little glass test tubes that the Tulip needles come in though.
I started with the Tulip sashiko needle. It’s enormous! It was like sewing with a harpoon. It sews well though and is a good choice for this thick cotton thread.
I also tried the Tulip silk needles, which are meant to be fine enough to glide through silk. I’m afraid it didn’t, really, even after a swipe with the emery. I tried this one on a scrap of fine silk sari ribbon covered with a bit of silk organza, using my finest silk thread (I think it’s Coats fine silk but can’t read the small print on the tiny label. My ageing eyesight frustrates me no end).
I also used a fine hand-dyed silk thread in shades of purple. I used the Tulip silk needle for one half and a Bohin embroidery needle for the other. I don’t think you can tell which is which, but the Bohin was easier to use.
I guess the moral here is that a good needle is a good needle, whatever that turns out to be for you. It’s useful to try a few different types just to see how they work.