My name is actually Joanna van Kool, but I gave up with the web page under this name ( kept refusing me) and went with my husband's. My mother was named Catharine Margaret Jenner Lister and, like you, she was told she was a descendant of Edward Jenner so hence the name. Like you, I also live in Australia (Sydney) and when we visited Berkeley Museum in 1997 I was told that I was not connected to Edward Jenner. Apparently a great many people have claimed descendancy who are not. We actually had, my uncle was told, the only extant portrait of Edward Jenner. It always hung in my grand parents' dining room and my uncle eventually gave it to the Jenner Museum where I fancy it now resides in the basement. All a bit of a mystery. I wrote a historical novel about some women forebears and did a lot of research, but sadly this did not cover the Jenner branch. I think a search with a genealogical guide (U3A or the like) would probably tell you all you need to know. If you'd like to contact me in Oz at some stage my email is firstname.lastname@example.org
and my phone number is 02 80127479. Happy to talk.