Not sure if its feasible, but if a walking route that avoided the A350 were possible, it would be much more attractive.
Good question ... but I don't see it practically.
North from Melksham station, you need to be alongside the railway - be it on Wiltshire Council's track, or a new path on Network Rail land.
A hundred yards(ish) north, my suggestion is to cut across 15 yards of waste ground and through the fence to Foundry close. You need that exit to reach housing, most retail businesses, restaurants, bus stop for Lacock which are close by that point.
Options / extras / alternatives
1.1. If you carried on alongside the railway, you'll be behind the building block on one side of Foundry Close and your next way out without knocking buildings down or replacing someone's home with a passageway is just before Leekes' big store, across their secured yard; from the front of their yard, walkers could walk across Leekes' car park (alongside tha A350) rather than the footpath beside the road itself. Another possibility, I suppose, would be for Leekes to have a back door and hotel guests walk through the store. But it ain't open 24 hours.
1.2. If you carry on further behind Leekes (but is there space??) to Dunch Lane, then you could walk down the lane to the back of the hotel. How you rise up to the lane would need looking at too. It all could be done (after all, the railway was once double and broad gauge) but I don't think we want to put a hurdle in the way of redoubling and that exit will be for Leekes and the hotel; also for walkers headed up to Beanacre and across Dunch Lane bridge to housing there.
Looking to walk 'back' from the A350 once you cross off the railway path ... three elements:
2.1. You could have people cross McDonalds car park, and exit through the back of their drive through - re-arrangement needed, not sure if it's practical, no gain.
2.2. Outside Foundry Close housing ... without taking gardens (well - tiny yards) away, nothing easy and even then just a few yards gained. With option 1.1, taking over a further home you could produce a walk through
2.3 It would be possible to walk in front of businesses and across Leekes' car park rather than alongside the A350.
I would be very seriously concerned at the adoption of any of those options except 2.3. And I would be very concerned at any ideas that double cross the railway or the A350.
Where it might get better ... the Melksham eastern bypass proposals (earmarked to be nationally financed later this decade) reduces this section of the A350 to a more local road (B road?) and that will/would clean up the walk.
More good news ... additional support for the original plan as it provides a walking route to the village of Beanacre - putting the residents there within the station walking catchment.
Edit to add - pictures:
The walk alongside the A350
Proposal map - red as suggested, mauve as looked at above
Behind Leekes - unlikely for a path