Laura Robson is the best hope.
But maybe you can't become English by immigration, just British. Interesting question. Can an immigrant to England be English? I guess not, because you can only immigrate to the UK. I find this conondrum more interesting than the thread title.
Being a Aussie she,ll be Desendant from a English convict as the Aussie accent comes from the London cockney accent ! On a technical term Andy Murray,s English and I am Desendant from a Italian . So let's stop all the technical crap and just call her British . Seeing as she grew up in London , I,de say English !