If you are going to use an oil,then i would recommend applying with either a trowel(they make them especially for this job,it has rounded corners so no risk of scratching),just pour some on the floor and spread thinly,or use a buffer with a white pad,then cover pad with a cotton cloth and go over again to remove excess,ideal for larger areas.
If you are going to use a hardwax oil(a mixture of oil and wax,oil soaks in,leaving a thin layer of hard wax on the surface,i`d recommend a satin finish,the matt doesn`t really suit walnut,but it`s your choice,applied with short pile roller in very thin,even coats(make sure you remove any excess as you go along,as it`s hard work once it`s dried)
Both give a nice finish,both can be refreshed and allow for repairs,but the hardwax oil will give better protection