If I have a bitcoin wallet on say Exodus, but something happens to the people that make Exodus, could I upload the 12 word private key on a different bitcoin hot wallet such as electrum and get my coins back?
The actual term is recovery seeds not private keys, it's combination of 12/18/24 words in common cases and as others pointed out it can be imported into any wallet that supports it. The process is pretty simple, just create create new wallet - standard wallet - I already have a seed then enter the 12 or 24 recovery seeds meanwhile don't forget to click the BIP39 box which choose the right derivation path /44'/0'/0'/0 for the seed you're entering.