So: What happens to the BTC stored on a hardware wallet like Ledger or TREZOR when these changes take place?
I don't understand how if it isn't connected the changes could apply to the BTC stored there, rather, only the system as a whole.
How could one wallet support one thing and another support a different thing? Wouldn't there have to be consensus in order to transfer/use coins for say, someone like me with stored BTC on a hardware wallet or cold storage hard drive?
For the Bitcoins stored on your wallet nothing changes. Just dont use yours Bitcoins after Aug 1 until the dust settles. There are possibilites:
1) BIP148 fork dies up OR BIP148 fork replaces Bitcoin because it gets most mining support: you will be able to use Bitcoins only from your wallet as usual
2) BIP148 fork lives forever but is minority chain thus altcoin: you will be able to use Bitcoins from your wallet as usual, but if you want use the BIP148 altcoin as well, you must first get more infomation how to do this before you ever start using your Bitcoins