Just a couple of updates. There was a second son that we didn't have a baptism for, and we concluded that he would have been born between Mary in 1801 and Jane in 1805, and perhaps didn't live long enough to be baptised. In fact he was baptised John and we found his burial:
John Simmons burial 28 Jan 1805 age 6 months son of John Simmons basket maker.
I got the death cert I mentioned, of my cousin's grandmother, from the GRO. A first for me. It was a very smooth operation and much quicker and cheaper than here in Australia, and it showed she died from acute appendicitis, general peronitis and cardiac failure. At age 30, less than two months after giving birth. Which certainly sounds childbirth-related.
I have spoken in detail to a member of the Thompson family who emigrated to Australia in 1951, completely oblivious to the fact that they were following in their ancestor's footsteps, although in very different circumstances. They were fascinated to hear the story and the woman I spoke to was grateful that I had taken the trouble to contact her, because she always though she had no relatives in Australia. The family had traced their story back as far as Elizabeth Burrell's daughter Elizabeth Symonds/Simmons, but the story had been a complete mystery beyond that.
I am continuing on with the second daughter, Mary, who had at least one daughter. The two boys died young. And the other two girls probably married and got lost under their married names. But I'll keep at it.
Thanks again for all your help.
Geoff