Something like a Class 380, from Siemens it depends on how much one is willing to pay them for these units, but the 380 would be more ideal for services around GM than most other units with their 23m long carriages, in 3 car formation.
Since it's the latest built and if the cheque book was thick enough, it wouldn't take long at all IMO to get then units built and over to the UK. End to end build time can be quite short if your wallet is thick enough, and if it's part of a much bigger order, it's much easier to exercise some cash to get a significant amount of units earlier rather than ordering a small amount for now, ordering a large amount with some of them now, then a continued delivery programme, agreed with the supplier when the contract is signed.
Either way, if the order was put in now, I reacon we could have enough units in place by the time the Chat Moss is open and wired up to Wigan, where an additional (in addition to the TPE units) 10 units would be plenty of capacity at the time of introduction, with significantly more being delivered in time for the route over to Liverpool being finished.
Of course a continued delivery programme would be in place so that the Class 323 units could be cascaded to LM to increase capacity on the cross city line with 380 units replacing these like for like in Manchester in either 3 or 4 car form. Maintained at a new depot built by Siemens under an operating contract at somewhere like Wigan.