Is it obligatory to slaughter the sheep vowed for the healing of the wife immediately on the upcoming wedding anniversary, or is it permissible to delay and slaughter them in parts over several years, and is it permissible to eat from them?

If your intention with the sacrifice is to draw closer to Allah, then you must fulfill the vow. If you specified a certain time for the vow, you must adhere to it; otherwise, you will be liable for an expiation for breaking an oath, while the obligation to fulfill the vow remains. However, if you intended the anniversary date in any year, you may fulfill it in any year, with preference given to expediting it.

As for the number of sacrifices, distributing them, sacrificing them all at once, or the one who vowed eating from it, all of that depends on your intention.

It is disliked to make a conditional vow. And if choosing the anniversary date for celebration and making it an Eid, this is not from the customs of Muslims, and it is better to hasten the fulfillment of the vow before this time, unless the date is suitable for reasons not intended for themselves.