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In my case, I withdrew from my exchange account from wallet ABC (if I look up the trx on block explorer, it says it came from public key ABC). A few weeks later I went to deposit more money into my exchange account, however I accidentally sent the bitcoin to wallet ABC instead of the nominated 'deposit address' on my account.
I'm freaking out that my coins may be lost. Would the exchange be able to sweep this wallet and credit it to my correct account?
Yes, it would be possible because they hold the private keys for that address. You will need to contact the exchanges customer support. However while they may be technically able to do so, they may not actually credit your account.
They probably already credited whoever's address that is. Say Alice sends bitcoins to her deposit address. When you withdraw, the exchange can use any bitcoins they have, which might include the ones Alice deposited. So your bitcoins will come from her deposit address. If you send bitcoins to that address, the exchange will credit Alice.
