In the key of C Major what is the Roman numeral for a B ♭/D ♭/F chord that goes to a flat VI?
Like in any Roman Numeral analysis, minor chords are represented by lower case letters and major chords are represented by upper case letters. In addition if the root is a lowered scale degree you put a flat in front of the scale degree and if the root is a raised scale degree you put a sharp in front of the scale degree.
So making a name for any major/minor chord with Roman Numerals is easy. If you were 100% sure the key is C Major you would have bvii and bVI as seen below.
X:1 L:1/2 M: K:C V:2 clef=treble "bvii"[_B _d f] "bVI"[_A c _e]|| %
The thing you need to be aware of is the Roman Numeral analysis is a tool that has its uses and its limits . As you get away from functional harmony concepts and ignore other harmonic concepts like secondary dominants and modulation the results aren't as useful as they could be. Even when it makes sense to use chords outside the key in the case of borrowing, too much can signify you need to look at the progression in a different light.
In your case if you were in the key of C, I doubt the you would still be in the key of C seeing the progression Bbm, Ab and if you have an Db next I would argue that you are in the key of Db major instead.
I agree that it sounds like a modulation. Determine the new key and analyze in terms of the new key and put brackets around all chords relating to the new key. If the new key is Db Maj, then Bb is probably minor and it is vi of Db, the new I, and so on.
It's bvii, bVI. It COULD be the beginning of a modulation, maybe to Db major (or several other places). But it doesn't have to be a modulation. If the next chord is V, or a climb back up to I, it could just be a touch of colour in C major.
My Understanding of the 'Roman Numeral Notation' depends on the Scale Degree. In the Major Scale of 'C', the 1, 4, 5, Are Capitalized. The 2,3 6, are lower case. The 7th, is also lower case, as presented... I ii, iii,IV,V,vi,vii. in the key of 'C' Major,'d' minor, 'e' minor, F Major, G Major, 'a, minor, 'b, diminished.