The "key" in many cases can be determined by the tonic note of the piece, and the tonic's position in the diatonic pattern.
It often happens that a modal music's key signature is written in a parallel major/minor key, and adjusted into the modal pattern by adding or removing accidentals.
I believe this is done so as not to confuse musicians that have learned only Major/Minor as key signatures and scale. Musicians that learn Major-centric theory are expecting only one of two tonics in a key signature, the Major tonic and the Minor tonic.
For example, you will see a Dorian tune (d-ef-g-a-bc-d) written in D minor key (one flat, the Bb in the key signature), and every single Bb in the piece will have a natural sign in front of it. The same is true for Mixolydian pieces, (g-a-bc-d-ef-g) where the F# is in the key signature indicating G Major, but all the #s are removed in the piece with accidentals.
In some method and instructional books, you will also see the key of C major, but the piece is actually G major, and the F#s are added in as accidentals. I believe this is usually because the method book hasn't advanced to explaining key signatures yet.
To answer your example, if your key signature is the same as C major and you see the Fs turned into F#s, if the tonic is C then you would say you are in the key of C Lydian, even if the key signature indicates otherwise. You would not say that you are playing C Lydian mode in the key of C Major (except to describe how the piece was written out in the transcription). If the key signature shows C, but the Tonic is G and F#s are added, then the diatonic pattern you are playing is G major.
Modal music will be described by its actual key, so you will hear "E Dorian", meaning the tonic is E and the scale pattern is Dorian. The key signature in this case should have two sharps in it.
Each Key signature contains the notes for all seven natural modes. The mode is determined on which tone in the pattern the melody or piece is focusing on (the tonic). Of the seven, Ionian is the same as Major Scale, Aeolean is the same as Minor Scale