These are the first two bars of J. S. Bach's Fughetta in c-moll (BWV 961) as edited by Henle.
Does the second trillo in the second bar start on B flat, so that it cancels the natural on the previous B? I cannot understand the second natural on the next B in that bar otherwise.
More generally, are ornaments supposed to ignore accidentals appearing in the bar and ‘see’ only the signature?
The general rule, given in Do accidentals earlier in the measure affect a trill?, is that preceding accidentals do affect the ornament.
This case, however, is an exception. It would be very odd, particularly for Bach, to trill between B and Ab, an augmented second. The ornament should be played with Bb. Every edition on IMSLP by a major publisher includes a Bb above the trill sign.
That ornament is a mordent not a trill, there exist an upper mordent and a lower mordent. A brief overview on it's execution can be summised by the following picture. Typically, unless otherwise notated the mordent would follow the key signature.
The notation of the B natural may indicate a passing chord in C minor. There may be the effect of a passing chord or it may simply be a lower auxiliary note. These passing notes often have the lower note raised by a semitone to give the effect of a resolving leading tone.
As with all things Bach the theory is ambiguous on purpose. It is one of the things that makes the textures rich and interesting.
