1. See if there is a release in the brake lever as this is sometimes where it is located.
2. Set the brakes up so the cable adjuster is screwed all the way out when the brakes are adjusted properly. Then you thread it in to gain clearance to remove a wheel.
3. Install the wheel with the tire deflated and inflate it in place.
4. There are in-line cable adjusters you can install to supliment the built in ones and gain even more slack.
5. Carry a wrench to loosen the cable clamping bolt when the wheel need to be removed or installed. Use the same wrench to tighten the cable after the flat is fixed.