There’s electrical interference to consider - the signalling on the West Coast is very different to that on the GE and different still to the MML. The 350s will have been “tuned” for lack of a better word to not interfere with the signalling on the West Coast, the 360s the GE and the MML. Those two “tunings” may be far from identical or compatible.
That electrical output has to be assessed as part of route compatibility. Sometimes it is just a tick in a box, often it involves subtle tweaks to train software and control electronics however.
Worth mentioning too that units/locos with DC drive (i.e. motors) are, on the whole, significantly less messy than those with AC drive. 317s have operated over all three of the GEML, WCML & MML, after all - with little to no change AFAIK.
88s and 92s (the latter infamously dirty in terms of interference) have also operated over the GEML (to Ipswich in the latter's case) and WCML (but not the MML).
Re the signalling, much of the WCML is now covered by axle counters (inc. Bletchley - Northampton), the GEML is entirely track-circuited, and so is the electrified stretch of the MML.
It could be that the 360s lack immunisation against axle counters - unsurprisingly, they've never been fitted with them as there's been no need to do so for the routes they operate on (past and present).
I'm also unsure whether the GEML/MML (or both) have Insulated Block Joints (IBJs) in the rails, i.e. whether the circuits are
single traction rail return (with IBJs) or
double traction rail return (without IBJs).