Is there a term for the group of non-natural notes, i.e. notes altered with sharps or flats?
I’m a french speaker, as such I usually call (double) sharps/flats “alterations” — which I’ve found in the Lilypond glossary, but seldom anywhere else. “Altered notes” thus seems perfectly reasonable to me.
1) On the one hand, you can denote which tones should be flattened or sharpened by default. This is typically done by writing those sharps and flats at the beginning of the staff, after the cleff, but before the time signature.
This is commonly referred to as "key signature". This is a bit of a misnomer since the key signature alone is typically not enough to deduce the key. Key signatures can appear halfway a staff too, but the principle is the same.
In relation to your question, the purpose of the key signature is to define which tones are "natural" to use your term; that is, it sets a standard for which sharps and flats are implied for the section to which the key signature applies, and it results in a more clear notation since those flats and sharps then don't need to be explicitly denoted.
2) Then there are accidentals. Accidentals are notes that are raised or lowered temporarily, deviating from that what is implied by the key signature. Note that this may be done either by sharpening or flattening a note, but it may also involve adding an explicit natural sign in case that tone was raised or sharpened by default, as implied by the key signature.
Accidentals might indicate tones that are outside of the current scale (or key, or both). However, that need not be the case. In the melodic minor key comes in two flavors: the ascending flavor has a major sixth and major seventh, whereas the descending one has a minor sixth and seventh.
Typically the major sixth and seventh need to be denoted by accidentals in order to raise them with respect to what is implied by the key signature.
For example, a key signature with one sharp may indicate E minor. The one "default" sharp in this case is f#. In such a piece you will often find accidental c# and d# (which are the major sixth and seventh of e respectively). And if the key signature would have, say, 3 flats to denote C minor (b flat, e flat and a flat), then you might find a natural a and b, which restore the all notes on a and b lines from their former "flattened" state (as indicated by the key signature) back to their natural sound.
3) Sometimes, accidentals are applied to indicate sounds that are outside the current scale. An example is chromaticism, which "fills" the gaps between diatonic scale degrees with semi-tones that are themselves not in the scale. The notes themselves are, to the best of my knowledge, still called "accidentals". But when you talk about the harmony that is implied by the cords that have these out-of-scale notes, I think then the term is to talk about "alterations"
For example, esp. in the minor mode you might encounter a so-called Neapolitan sixth chord. This is the first inversion major triad on the minor 2nd degree. Let's say you're in A minor (key signature with no flats, no sharps), then a first inversion major Bb chord is the Neapolitan sixth. The root of this chord is Bb, and this is referred to as an "alteration". The in-scale chord on the 2nd degree would be the diminished triad b-d-f, and by alteration, which diminishes the major 2nd to a minor 2nd, and the chord becomes a major triad bflat-d-f.
So as a chord note, the b flat is called an alteration, but the denotion of the b-flat is simply an accidental.
when the context is the major key, a note that is not part of the key is called a non-diatonic note.
In my experience (playing mostly early music in the UK) they're called 'chromatic' notes. This is from the term 'chromaticism', and the idea that throwing in the occasional extra sharpened or flattened note adds 'colour' to the music. The chromatic scale, of course, contains all these possible notes.
Note that this not the same as a piece in a minor key which notates the necessary sharpened or flattened sixths and sevenths as accidentals (because they're part of the minor scale and so aren't chromatics), or a piece which changes key without changing the key signature in the notation.