All the documents relating to the study are here:
https://www.eastsussex.gov.uk/roadsandtransport/roads/roadschemes/rail/
The trackbed is either a farm track, bridleway or railway for most of the length between the Uckfield Bypass and the former Hamsey junction. But the Lavender line is an important tourist asset, and Barcombe mills station is a private dwelling which has filled in the trackbed between the platforms. At Uckfield the trackbed is obstructed by the Bypass, and partially by a sewage farm and industrial estate. At Lewes the original route through the town is almost totally obliterated, so NR proposed using the original alignment via Hamsey, which is intact (but involves 2 level crossings or a realignment). This means a reversal beyond Lewes is necessary to reach Brighton. There's also the small matter of Isfield level crossing, which not only would cut the village off from the main road every time it closed but is very close to a T-Junction so risk mitigation would be very high (if allowed at all).
Not insurmountable obstacles, but not exactly a 'ready to go' project.
BML2, while a worthy suggestion, hinges on a 1.5 mile tunnel through chalk downland in the middle of a National Park. I imagine there have already been several threads on this forum about potential Lewes-Uckfield and 2nd Brighton-London connections so won't go off on one here.
@SussexLad, give your username you really should be spelling Lewes correctly!