mark99
The Home Office will always require the name in the passport.
The ILR application form will ask for name as in passport.
Being married legally allows the woman to use either her maiden name or to take the family name of the husband or a combination.
The marriage certification is the legal link and legal confirmation of the name change.
So your wife can use whatever name(s) she would choose.
Just as example, my wife used her maiden name as her new middle name. Not just as Philippine tradition but also to keep a nice link between all names also.
On a practical level, it is advisable to maintain consistency of ID. Means always use the same one. Less complication the better.
Hope that helps you.