I'm getting quite a lot of search hits for "mastercard atm bishkek" so I thought I would post something useful about this topic.
The only bank in Bishkek from which you can withdraw cash on a Mastercard is KazKom (Казкоммерцбанк), the Kazakh bank here. There are several ATMs in the following locations:
1. Tzum, on the outside of the building facing Chui. There's also one inside, on the first floor, at the very back.
2. The main KazKom branch is on the corner of Sovietskaya and Bokonbaeva. There are several ATMs there, and some of the staff speak English.
3. Dordoi Plaza. First floor, central aisle, there are a couple of ATMs. One is KazKom.
4. Along Chui, headed west, just past Ala-Too Square, there is a very small machine in a wall. You have to look closely to see that it's KazKom. It's before all the travel agencies.
There may be other locations, but I've never found them. It took several months for me to accumulate all that knowledge, and I hope that it may prove useful to someone, somewhere! Generally, I wouldn't advise travelling in Central Asia with a Mastercard. Visa is much more widespread here.