Most large removal vans have three seats in the front, with three employees occupying them. So it's simply not possible to accommodate customers in the vans. (I travelled in the van with the driver when I moved from Sheffield to Suffolk but that was only possible because I was using a 'man with a van' service, rather than a proper removals firm, to shift my relatively few belongings and small amount of furniture).
Pickfords, for example, provide advice on planning your journey but they don't offer any direct assistance with customers own travel:
https://www.pickfords.co.uk/moving-to-the-other-side-of-the-uk
CrossCountry trains, on their Edinburgh to Plymouth route, run directly between Sheffield and Exeter (if that's where you've finally decided to move to), with Advance tickets available from £59.40 per person. (That fare uses a sample date of Thursday 2 September, leaving Sheffield at 0956 and arriving at Exeter St David's at 1444).