May as well trot out one of my regulars again:
BML2 doesn't make sense as a way to provide capacity to London. Going via Uckfield simply takes too long. Reinstating Uckfield to Lewes for regional trips though? Great idea. But that's not going to help the BML.
My suggestion would be to do three things - 2 major interventions and 1 minor one. First, a minor project worth doing regardless - build a loop south of Lewes station so services from the north can run through Lewes to Brighton without a reversal:
Secondly, quad track from Keymer Junction to Haywards Heath station. Means a new short (227m) tunnel bore and widening a short (70m) viaduct, but other than that it's pretty straightforward stuff. Network rail are already planning to potentially quad track through Wivelsfield as part of the grade separation of Keymer Junction - this just extends the quad track to Haywards Heath.
The final, and most contentious bit - extend the quad track north from Haywards Heath to Copyhold Junction, rebuild this with grade separation and rebuild the line to East Grinstead via Horstead Keynes, compensating the displaced Bluebell Railway appropriately.
Based on the old line speeds that would give a route that takes roughly 5 minutes longer than the current BML, and importantly, has an interchange with the current line at Haywards Heath, providing connections to Gatwick et al. Merging the via Lewes and East Grinstead services means you can combine their paths north of South Croydon, meaning that you can continue to use the existing routes more effectively, meaning little to no expensive London infrastructure.
...and by rebuilding Uckfield-Lewes as a basic diesel railway (after all, if you don't dress it up as a London mainline, you don't need electrification on day 1) you keep the costs low enough to justify getting the rails put back and you just extend the Uckfield services to Brighton, Seaford or Eastbourne. Add on the electrification bells later.