Will this affect the BTC wallets just like BCH?
Yes, Bittrex just began to replace BTC by Bitcoin Gold. First of all it was airdrops but this the beginning only. We will see all its fork create a chaos for ordinary traders, investors and other people working in cryptocurrency.
What do you mean replacement of BTC by BTG? Please explain. There is no such replacement on bittrex.
They will credited or already have creditd your BTG to your account, that's all. They're not changing anything about BTC on the platform.