Quickest change point detection is concerned with the detection of statistical change(s) in sequences while minimizing the detection delay subject to false alarm constraints. In this paper, the problem of change point detection is studied when the decision maker only has access to compressive measurements. First, an expression for the average detection delay of Shiryaev's procedure with compressive measurements is derived in the asymptotic regime where the probability of false alarm goes to zero. Second, the dependence of the delay on the compression ratio and the signal to noise ratio is explicitly quantified. The ratio of delays with and without compression is studied under various sensing matrix constructions, including Gaussian ensembles and random projections. For a target ratio of the delays after and before compression, a sufficient condition on the number of measurements required to meet this objective with prespecified probability is derived.