Is it necessary to perform ghusl immediately with the water used for wudu’ by the one who cast the evil eye, or can it be delayed, and is there a specific period after which the effect of his wudu’ water dissipates?

We do not know of any scholar who has stated that it is obligatory to perform a ghusl with the wudu' water of the 'ain (person with the evil eye) immediately or within a specific timeframe. It appears that performing ghusl with it fulfills the requirement. The 'ain must respond to the request of the ma'yūn (person afflicted by the evil eye) to perform wudu' for them. However, the ma'yūn is not obligated to request this, and can suffice with ruqya, adhkar, protective measures, and reliance on Allah.