Nonintersecting Brownian bridges on the unit circle form a determinantal stochastic process exhibiting random matrix statistics for large numbers of walkers. We investigate the effect of adding a drift term to walkers on the circle conditioned to start and end at the same position. For each return time $T<π^2$ we show that if the absolute value of the drift is less than a critical value then the expected total winding number is asymptotically zero. In addition, we compute the asymptotic distribution of total winding numbers in the double-scaling regime in which the expected total winding is finite. The method of proof is Riemann--Hilbert analysis of a certain family of discrete orthogonal polynomials with varying complex exponential weights. This is the first asymptotic analysis of such a class of polynomials. We determine asymptotic formulas and demonstrate the emergence of a second band of zeros by a mechanism not previously seen for discrete orthogonal polynomials with real weights.