From Shenfield the train journey to SHSB would be the easiest - Shenfield to Prittlewell arriving about 0750 then about 15 minutes' walk. No idea about the contract buses to either school (we are local) so there may be one you would feel happier with but personally I prefer public transport for its flexibility. Also, for the train, you can get a discounted Scholars' season ticket (either termly or a more advantageous annual - see http://www.greenrailtravel.co.uk/acatal ... ation.html
Shenfield to Chalkwell for WHSB is more complicated as they are on different systems; you could do Shenfield - Southend Victoria (a few minutes on from Prittlewell) then get Arriva Bus 29 or First Bus 21 back westwards, I suppose. Both these routes stop directly outside WHSB. You can add PlusBus to your Scholars' season
Lots of DC use the train, and the local buses are pretty full with kids from all the local schools at "coming and going" times.