I'm also moving (next week actually) and I bought these
Ambassador Packing Carton Double Wall Strong Flat-packed 457x305x305mm Ref SC-64 [Pack of 15]: Amazon.co.uk: Office Products
They're a great size that even when filled to the brim with books you can still lift them. Not so big that they're difficult to carry and they're very strong (I have them stacked 6 high in my storage unit without a problem at all). At about ÂŁ1 each and delivered next day for nothing if you have Prime they are a steal.
I also bought this
Einhell EINBTHT90 Folding Truck 90 kg Capacity: Amazon.co.uk: DIY & Tools
which has saved my back from a lot of pain lifting/moving the boxes. It's been very robust so far and definitely worth the ÂŁ25. There was this one which looked better
Sack truck Hand trolley with foldable toe - Pneumatic tires - Heavy duty 200kg/440lb: Amazon.co.uk: DIY & Tools
but they couldn't guarantee to get it to me in time before I moved. Still, very happy with the one I did get and even if it broke now I've had my ÂŁ25 worth.
If I'm finished with all mine by June you're welcome to have all my boxes for a nominal fee as I'm never ever ever moving again...
G