We are attempting to open and close a circuit (an on/off switch) wirelessly via a mechanical button. We'd like to use an arduino uno, a Bluetooth shield and receiver, and a mechanical button.. is there a way to do this? How should we proceed ?
Yes it's possible.
If you really want to use Bluetooth (why?) then I would imagine you'd need two Arduinos and two Bluetooth shields (one pair for the transmit side, one pair for the receiver). You'll also need a relay module on the receiver to do the switching.
The button will be attached to the "transmit" Arduino. When the button is pressed it will signal it to the receiver, which will operate the relay.
Google "Arduino bluetooth control" and see what you find - each part of this project has been done, and you should be able to piece together the various parts.
If you encounter specific problems, come and ask and someone will try to help.
If you removed "bluetooth" from the requirements, I'd suggest using ESP8266s, which have wifi built in. That would save you a lot of complexity. Again, hunt around on Google to find other projects which implement the various parts of this, which you can piece together.