Egypt -at least according to nasser- never looked at the war in Yemen as a sunni struggle, its a very naive orientalist thought anyhow. here is Nasser point of view its clearly a conflict between Socialist Secularism and Islamism.

what extent was an increasingly hardline vision of Islam a tool by which Saudi Arabia could draw people and states away from Egypt's ideology

that is a very complex question , a good start would be to understand what is saudi role in the arab world, how capitalism/money played a role on gaining influence

Wahhabism was largely confined to the Arabian peninsula until the 1960s, when the Saudi monarchy gave refuge to radical members of the Muslim Brotherhood fleeing persecution in Nasser's Egypt. A cross-fertilization of sorts occurred between the atavistic but isolated Wahhabi creed of the Saudi religious establishment and the Salafi jihadist teachings of Sayyid Qutb, who denounced secular Arab rulers as unbelievers and legitimate targets of holy war (jihad). "It was the synthesis of the twain-Wahhabi social and cultural conservatism, and Qutbist political radicalism- that produced the militant variety of Wahhabist political Islam that eventually (produced) al-Qaeda>

[here is the full study by curtin winsor on that subject] (http://annaqed.com/en/terrorism/saudi-arabia-wahhabism-and-the-spread-of-sunni-theofascism)