Very little:
WW1's P14 was used on a limited basis as a sniper rifle in WW2, more often relegated to Homeguard duties. Not a significant player.
A lot of the No. 1 mk. III* rifles of WW1 vintage were refurbished and used again. No change.
Several arsenals (Lithgow, Ishapore, BSA) still made No. 1 mk. III-pattern rifles because they had spare parts, tooling, and EVERYONE needed rifles.
No. 4-pattern rifles were fundamentally the same but logistically entirely different; like 8 parts were interchangeable. The rifle had aperture sights instead of leaf, tweaks to the barrel and receiver,and the front foresight and appearance were changed entirely. Basically they were a slight upgrade in accuracy and significant reduction in manufacturing effort.
The No. 5 Mk. I was a lightened carbine version of the No. 4 mk. I
For details and pics look at my submission history.