I did the same move a couple of years ago. Please don't underestimate the size of the task.
The removal companies I spoke to wouldn't touch the machine tools – they are too big and unwieldy for them. What I did was to hire 20 'croc boxes' – those tough plastic crates with interlocking lids. They held all the chucks, tooling etc. Cardboard boxes aren't strong enough. What I found after carefully packing a couple of croc boxes was that became too heavy to lift – you can only really put a couple of chucks etc per box. Put any oils into plastic bags or use clingfilm – any leaks in the van could wreck whatever is lying underneath.
Start packing the workshop early as I found it takes ages and I had far more stuff than I realised. I used the opportunity to sell some stuff as well.
I too used Steve Cox to shift my lathe and milling machine and shaper – he is very safe and professional – but make sure he knows exactly what to expect at either end
He does offer a storage service as well AFAIK so he could take your machines a few days before the move.
It is quite awkward to manage because you probably have to move the workshop and house contents on the same day and coordinate both and you have heavy machine tools being moved in at the same time as your house contents – it is stressful and I was absolutely knackered at the end of the move day.
I found the removal guys to be amazing and they worked none stop for hours – I made sure to keep them supplied with tea and biscuits and to tip them generously at the end.
Andy