Everyone I've spoken to about chord progression, including rock musicians and college professors, pronounces the Roman numeral name. So, I is the "one chord", ii is the "two chord", etc. etc.
If the chord is not the strict diatonic chord of the key signature some modifier will probably be added. So, if the key were minor, but the final chord is a major I, that might be called something like the "major one chord" or the "major tonic chord".
V/V is usually called the "five of five chord".
Chords on the seventh scale degree come in various qualities and the exact pitch of the seventh scale degree varies in minor keys so the exact spoken description of those chord is a bit fluid and sometimes assumes context. viio might be called simply the "seven chord", or it may be called something like the "diminished leading tone triad", etc. etc. In minor keys VII might be called simply the "seven chord" or the "flat seven chord". That is a case where context often assumes the specific root pitch and chord quality.
When dealing with common progressions like ii V I or I6/4 V I, you might hear "two, five, one progression" or "cadential six four, five, one progress" or something similar.
Basically, just say the name of the Roman numeral, and if anything is either not strictly diatonic, or otherwise chromatic, or potentially ambiguous, use enough language to be clear. I'm thinking of potential confusing things like a "augmented six chord" - being an augmented triad built on the lowered VI minor scale degree - and and "augmented sixth chord" - which is a different chord with the lowered VI minor scale degree in the bass and an augmented sixth above it. Neither chord is strictly diatonic and so will probably need a fairly wordy description to be clear.
For what it's worth, I would not use solfege syllables - DO RE ME... to name chords/chord roots. Solfege make me think more about melodic aspects whether that melodic voice is the bass or any upper voice.