Two carrier lists
A posted Wholesale Cartons parcel, for senders of fewer than 1,000 parcels a year, uses the Royal Mail small parcel maximum of 450 x 350 x 160 mm on the outside, in force as one size since October 2014. The older pair was 450 x 350 x 80 mm and 350 x 250 x 160 mm.
Amazon is a different list. The Amazon category publishes FBA small parcel at 35 x 25 x 12 cm and Amazon Shipping small parcel at 45 x 35 x 16 cm. A carton that fits one can miss the other.
Crash lock and rolls
Crash lock cartons are described as 150K/T in six sizes that fit Royal Mail small parcels. A tube, if you send one, uses the roll exception of 450 mm by 80 mm diameter.
A small parcel costs less than a standard parcel on the Royal Mail notes. That sentence is not an Amazon rate.