I've just ordered a new 5870 and I will soon be selling my 560Ti and buying another 5870.

For the time being, is it possible to use both the ATi and the NVidia card?

If so, how will I go about doing it? Is it just a matter of running two instances of a miner, but changing the flags?

Yes, poclbm has the platform flag, so you can select between nvidia, amd, or intel platforms.