To the OP

I saw the organiser of the Walthamstow Wassailers and she is not aware of any Essex specific wassails.

Somone I haven't seen for a few years but used to be a regular drop in at the club is Tony Kendall "The Essex Man". Here is a link to a website which gives more info and refers to Essex Wassails but Tony not only collected but wrote "folk" related songs on Essex themes so they may be his own work.