What you need to establish first is which hospital you are attending.
The “Barts Health” NHS Trust encompasses six hospitals including St Bartholomew’s Hospital and The Royal London Hospital in Whitechapel. St Bartholomew’s (popularly known as “Barts”) Hospital is not at Whitechapel but is very close to Smithfield Market (in fact, about half way between Smithfield and the Old Bailey). If it really is Bart's you want to go to then Ramsgate to London Bridge is the way to go. Change there (same platform, No 6) to a St. Albans/Luton/Bedford bound train for three stops to Farringdon. It is a short walk from there (about 5 mins) to Bart’s. Anybody will direct you via Cowcross Street, through the market and down Giltspur Street.
If it is the Royal London Hospital you are attending then mushroom’s route is good enough. Personally I would travel from Ramsgate to London Bridge, walk across to Borough High Street (south side of London Bridge itself). From there catch a No 40 bus to St Botolph Street and from there you can catch a No 25, 205 or 254 to the Royal London. The reason I suggest this alternative is because some parts of the Overground are former Underground routes and the Shoreditch-Whitechapel stretch is one of them. It is not “deep level” but I don’t know what your fear of the tube system involves.