What to do with wallets when the fork happens

If I understand it correctly, when the fork happens there will be two different chains that compete for legitimacy, with the victor for the title of being called 'Ethereum' (ETH) pre-decided as going with EIP-1559. The 'loser' is going to get a new designator (say Ethereum Medieval, rather than Classic).

But my wallet exists prior to that fork. So if I understand it correctly, a fork will in effect cause me to have two wallets — one valid on one chain and invalid on the other, and vice versa. E.g., I will have X amount of ETH and the equivalent X amount of ETM*. I imagine it would make no sense for the private or public keys or address to change with the fork.

Here's what I want to know, though:

• What do you think is wise to do with your secondary wallet? Keep both wallets until the market settles down? Or are you going to sell the contents of one of your wallets immediately?
• Major exchanges support the current forks (BCH, ETC). How long does it usually take for them to accept the alternative after a given fork? For those seasoned currency hodlers, what do you remember from when ETH and ETC split?
• I primarily use exchanges for acquiring ETH. I can't exchange from my wallet unless I use an exchange, or find another way to sell my ether from the wallet. How do you plan to exchange your ether from your wallet(s)? I am leery about using channels that aren't an exchange like Kraken, Coinbase, or the like. Are there good ways to go about this?
• Similar question for wallets. If I have a trezor or some software wallet, I am not sure I know how they'd recognize the addition of wallets from the fork. I am pretty tech savvy, but not savvy on this tech angle.

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* Or whatever the 3-letter designator becomes. You know what I mean.