Thank you for your kind words, Maria, but I can't claim to have any special research skills and there are many others here on Rootschat more knowledgeable and experienced than I. I've 'learned my way around' the records in counties where I have ancestors, joined family history societies and so on, and have learned from and been helped by other Rootschatters. Quite apart from that, we all enjoy the challenge of a good mystery!
Many small schools were set up by individual schoolmasters/mistresses and came and went fairly quickly. The Dorset History Centre
https://www.dorsetforyou.com/dorsethistorycentre or Poole Museum
http://www.boroughofpoole.com/leisure-and-culture/museums-and-local-history/museum/may be able to guide you in the right direction with regard to the school, any major epidemics etc.
Both Timothy and Richard died after the registration of births, marriages and deaths started in 1837 so you can order copies of their death certificates, which will show the cause of death, from GRO. The cost, £9.25 per certificate, includes postage to anywhere in the world. If you decide to order them, you will need to quote the full page and volume numbers for the register entries you have already found.
It is such a shame that Timothy didn't last another ten years as his place of birth would have been shown in the 1851 census! Sometimes people are missing from parish registers simply because the minister forgot to enter them. The registers that we are familiar with are monthly summaries of the services that took place. Some ministers were scrupulously accurate and entered additional details (such as dates of birth with baptisms) whilst others were appallingly sloppy and got the details wrong or left out entries altogether. One local vicar, looking after several rural parishes, entered my Alner family as Allenor, Alver, Allen and Arnold. I had to check the original parish registers then cross check that other families of those names weren't living in the area to make sure I had the right family members as all the other biographical details matched. I've seen marriage registers with entries completed then names crossed out and different names written over the top, dates scribbled out and changed, entries left half finished and gaps left, presumably for things the minister couldn't quite remember at the time.
Anyway, more information about the Oakes may come to light so we'll keep them on the back burner.