digital IIR filter delay and oversampling
Note that a given digital filter impulse response $h[n]$ will have a corresponding equivalent analog frequency response $H(e^{j\omega})$ through the sampling relations; hence with the given sampling rate Fs.

This means that if you change the sampling rate of the input of this filter, then the effective analog filter response will also be changed, probably that's not what you want to have. So you have to redesign the digital filter to recreate the desired analog filter response.

In particular, when you increase the sampling rate by two, then the digital filter $h[n]$ should be redesigned such that its digital frequency response cutoff frequency is halved, so that the associated analog cutoff frequency would remain the same. 

That means that, since the digital cutoff frequency is halved, the filter impulse response is doubled in length, the delay is doubled too. But don't worry, as the sampling period is also halved, the associated analog delay remains the same.

Of course, delay here refers to that associated with the decay time of the digital IIR impulse response $h[n]$.