From what I've read, it didn't really. For the most part the main belligerents of the Second World War began the war using the same grenade designs they had been using at the end of the First World War with the only main differences being simplified production methods.

Examples of this are the British Mills bomb, which was the staple British hand grenade of the First World War and remained in service in varying forms until the seventies.

The well known US Mk II 'Pineapple' grenade had entered service with US forces during the First World War and remained in service throughout the Second World War. Similarly, the Stielhandgranate or German 'stick grenade' reminded in service with German forces throughout both world wars with slight variations, notably the ability to detach the explosive head from the wooden handle for a greater variety of uses.

It was only once the grenades started being used during the Second World War that changes started to be made as soldiers discovered that grenades intended for use during a largely static, entrenched conflict were not quite suited for the more mobile conflict that the Second World War became. Most notably was the change in fuse time for British grenades when it was discovered that when German soldiers weren't confined to trenches, the rather long fuse gave enemy soldiers time to escape the grenades detonation. For this reason the fuse was shortest to four seconds.

Source: Firepower: The British Army Weapons and Theories of War1904-1945 by Shelford Bidwell and Dominick Graham

As below, the grenade first became important in World War One so the grenades in use were mostly newly-designed for that war and were fairly up to date, as weapons go.

During World War II the US developed the 'Beano', a grenade designed to emulate the baseball in size and shape. The designers went for an impact, rather than time, fuse and getting it right proved very difficult, and in the end it was scrapped.

When the British were desperate in 1940 they developed several low-tech grenades, including a glass bottle containing phosphorous and benzine, and the 'gammon bomb', basically a bag of explosive and later popular with irregular troops.

One new 'grenade' was the shaped charges used for anti-tank work, especially by the Germans and Russians. But these were deemed pretty suicidal and most nations worked on propelling the shaped charge via rocket or projector, leading to a whole family of weapons like the Bazooka and Panzerfaust.