Hi Alan. Elias Rumsey was indeed from Appleshaw, Hampshire. I am descended from Elias, through his son Henry Rumsey. Henry's daughter Sarah Rumsey emigrated to Dunedin, New Zealand in 1862. Her cousin Elias Henry Rumsey also emigrated to Dunedin around the same time, but via the Australian Goldfields.
I hope you are descended from the family also and look forward to hearing from you. Please send me a PM.
Peter Grant (London)