There are two types of oils used on wooden instruments: drying and non- drying. Almond oil, olive oil, peanut oil and most other edible vegetable oils are non-drying. Linseed and tung oils oxidise and harden. I prefer drying oils. You can varnish a table very effectively with a drying oil if you spend enough time doing it, and the result will be waterproof and more heatproof than most varnishes. After many years of using linseed on my instruments I have decided that tung oil is better. As an indication of the value of tung oil, the wooden work surfaces in my kitchen have been finished with pure tung oil and nothing else, and there has been no water penetration around the sink. My dining table is also finished with tung oil and I am happy to put a hot mug on it without fear of leaving a mark.
There are two types of linseed: boiled and raw. Boiled oil will harden faster than raw, but boiled oil usually has dryers added that may be toxic, and should therefore be avoided. (You lick your lips, so you should be careful what gets on them.)
African Blackwood (known in the USA as Grenadilla), the material used for making most woodwind instruments these days, is too dense for oils to penetrate. The oil will fill the surface pores, and that's about it. You can soak a piece of African Blackwood in oil for years and find perfectly dry wood just below the surface.
The next statement might cause an argument: I have seen no evidence that oiling an African Blackwood instrument will prevent it cracking. I oil my wooden flute because it makes it sound better, and I know it needs oil when it stops responding as it should. I have always preferred drying oils. Until recently I used raw linseed oil, applied quite hot (to make it thinner and more runny). For the past few years I have used tung oil, which is better doesn't stink like linseed. Non-drying oils like almond oil are only effective for a very short while, as they get wiped off when the instrument is swabbed.
I treat my headjoint with tung oil frequently, inside and out, and particularly in the embouchure hole. The body of a wooden instrument should only be oiled with the keys removed to avoid getting oil on the pads, and it should be left for a few days for the oil to harden. In practice, this usually means the body is never oiled. Boehm, incidentally, said never to oil the body.
Before you run off and pour tung oil into your instruments please do consult the maker, as other makers have different ideas. It is worth pointing out that museum curators have had many arguments about oiling old instruments, and this is an area of some conflict. In my case, as a maker, I tell the people who buy my instruments to use tung oil in *very* thin layers. Lots of thin layers that have had time to dry and harden will work better than a thick layer that has turned into a jelly with a skin.
If you do oil your instrument, use the oil very sparingly and wipe off any excess immediately. You need very, very little oil for it to do its job effectively.