Taking weekday station entry counts (available on TfL's Open Data site), the number of entries on the Epping Branch (all stations between Snaresbrook and Upping) is 42,460 per weekday. By comparison , the Hainault branch (Roding Valley to Wanstead inclusive) has 43,441 per weekday. This splits the demand 50.5% to the Hainault branch and only 49.5% yto the Epping branch. Using 'entries' data includes trips that do not use the line west of Leytonstone (previously I referred to the load across the link on each branch closest to Leytonstone, which is also available on TfL Open Data).
If you exclude Roding Valley, Chigwell and Grange Hill from both branches as they could go either way, the Hainault branch total falls to 40,517 entries per day. This does make Epping the busiest - but only just, giving it 51.2% of the combined total...