Oiling is a bit of a religious issue.

Some recorders don't need it because they've been pre-impregnated with wax. (Not as good an idea as it seemed at the time, and less often used these days). Some players don't oil at all, it doesn't make a huge difference.

The point of the oil is (1) to stop rapid variations in humidy level in the wood and (2) to improve the smoothness of the bore. Almond or grape oil work, mineral baby oil doesn't, linseed oil is weird stuff you have to understand better than I do to make use of it. Use a scrap of silk dampened with oil and push it through the bore with a kitting needle, or just twist and shove. You don't need a lot and you DO NOT want to get oil in the windway or on the labium (the sharpened edge that makes the sound) - that will lead to water droplets forming and badly buggering up the sound. Remember you can always add more, but you can't take it away if you've overdone it. Leave the instrument for a day or two without playing after oiling to let the oil soak in.

More important aspects to recorder care are not chew the mouthpiece and not to blow too hard (don't try to the clear a blocked windway by blowing, either shake or suck). And cut your left thumbnail short - I keep a pair of folding nail scissors on my keyring specifically for this. Also, never put anything sharp down the windway or near the labium, and don't store the recorder near a heater.

The instrument of mine where oiling made the largest difference was a wooden Rumanian whistle. It wasn't one of the usual crappy pokerwork type, and obviously had potential, but it was pitched in C sharp. The bore was very rough, so I sanded it out with fine sandpaper glued to a dowel and then oiled it. The smoother bore brought it up to regular D pitch.