Although there has never been such an Act of Parliament for the construction of the West Coast Main Line (it being a colloquial term to describe the network of various railways that make up the route from London Euston to Inverness, Glasgow Central, Edinburgh Waverley, etc), the bits you are looking for at Carlisle Citadel are the Caledonian Railway (the bit marked 'to Scotland"), and the Lancaster & Carlisle Railway (the bit that is marked "to Upperby").
Hope that helps.
The RCH Junction Diagram (second map down) may help!
Strictly speaking the main lines of the West Coast Route (when did WCML come into use?)are the top pair in from Upperby and the obvious bottom pair towards Scotland. All the rest marked Scotland are goods lines.
It's a silly name anyway. The west coast is only visible from the WCML for a short distance around Hest Bank and probably also around Kingmoor if you squint a bit. The sections where it's even the nearest railway line to the west coast aren't much more than that. And the ECML tag is only slightly more appropriate.