They will follow the franchise agreement to the letter as they won't run any trains they don't have to but the franchise agreement specifies the following for Monday to Friday:
https://www.gov.uk/government/public...tion-to-tender
First and last services depart London. First (no later than 5:45) Last (no earlier than 00:45).
I can't find the times for Southend to London so either it hasn't changed or I have missed it but there seems to be some attachments that are either missing or labelled incorrectly