My proposal I keep totting out when Uckfield to Lewes crops up is that whilst I agree it's worth building on a local case for connectivity, I think the only strong case would not be as a route to London, but as a route to Tunbridge Wells.
If relieving the BML is your objective, you need to interchange with it as some passengers will still want access to the stations along it. Routing them in the opposite direction over a much longer route will not really help them much.
However - if you instead add a new bore to the short (249yd?) Haywards Heath tunnel and double up the short (150yd?) viaduct south of there, then you get a segregated route all the way from Lewes to Haywards Heath via Keymer Junction. North of Haywards Heath it should be trivial to continue with four tracks all the way to Copyhold Junction, and thence restoring the route via Horstead Keynes to East Grinstead gets you a segregated route from Lewes to London that is only 7 miles longer than via the BML, but offers an interchange at Haywards Heath so you retain access to Gatwick, Three Bridges et al. without consuming paths on the BML proper. The ideal would be for Keymer and Copyhold to be grade-separated so you could even have cross-platform interchange at Haywards Heath.
Unfortunate for the Bluebell, but would be immensely beneficial for the region to have those links on the NR network again with all the associated frequent through services.
...and helpfully, it would also be a small start on the very-long-term objective of finishing the quad-tracking of the BML
