Well, the biggest problem is that the lines are paired by use west of Shenfield. Short of shutting down the whole line from Ilford to Shenfield, reversing the down main and up electric and turning the whole lot into a paired by direction route, there is not much we can do about that. Even if we just do the Shenfield-Chelmsford section, traffic onto the up electric will still have to cross the down fast. It might be possible to send the up electric parallel with the Southend loop at Shenfield, swinging the down electric out wide to avoid it, and there does seem to be room. However, it seems incredibly complicated, from Stratford, the down main is on the left, then second from the right, then would be second from the left. The up electric would go under a diveunder, then over a flyover, switching sides twice. Better just keep the lines paired by use.
Building separate main lines also means being able to avoid modifying stations. At Ingatestone, the station buildings are on the down side, so leave them as they are. They could slip the new lines between the old ones and the houses on the up side, or swing them 300 yards to the south and avoid the houses altogether. The first might allow new platforms, while the avoiding line would not. At Hatfield Peverall, the new lines would either have to scythe through the car park, or separate and follow the A12 for a bit. Whitham could be accommodated by slewing and reversing the direction of the current main lines (down loop becomes down electric, down main becomes up electric, up main becomes down main, up loop becomes up main). Kelvedon presents the same problems at Hatfield Peverall, only there the A12 has some tight curves to negotiate, making the avoiding line harder to build. Marks Tey again presents the same problem, and scything through the car park seems the only option there because the A12 is harder to access. Did there used to be a station at Lexden Heath? There is room for one. The proposed flyover would be a quarter-mile west of Colchester South Jcn, connecting the up passenger loop (extended and slewed wide) to the current up main (to become the up electric) allowing the new mains to pass underneath. As an alternative to moving all the buildings, it might just be possible to live with the current three through lines, but that puts a lot of pressure on the up main. A second line through the eastern diveunder, allowing freightliners from Ipswich to use it to access the down loop, might work to get them to the up electric without blocking up a platform. Without the flyover, that would be the only option. Only thing is, it does wipe out a road called Havering Close and flattens some of the gardens on the north side of Norfolk Crescent.
That only leaves Chelmsford, and the question is how to run the two halves together. Simply going around the side is not an option, we need four platforms. Going down to double track would mean up freightliners crossing the down main twice, and two flyovers would be a bit pointless. Swapping to paired by direction there would push up costs and make construction harder. The only real answer is to pull the whole lot down and start again. Two new lines alongside the old ones, new viaduct, new buildings (probably buried underneath), extra decks on the car park, the lot.
This approach is basically building a new double-track railway parallel to the old one. Although it means buying more land, it does not disrupt passing traffic, so means fewer closures. It is also easier to build avoiding lines than try to force through parallel to old ones at established stations.
Wow quite an essay I've written there aha. 
