Wallet to wallet in congested periods I have waited for nearly 24 hours for my BTC transfer to my new wallet.

Does the amount I sent fluctuate with the markets?

For example, If I send .01 BTC and while the transfer is in progress, the value of BTC goes up, will I still receive the profit when it arrives to me?
You sent X BTC, and you will eventually receive X BTC.

There is no profit or market condition involved here. You hold the same amount of BTC before and after.

You can of course choose to sell those BTC, and if the market value goes up, then yes you can sell at a higher price when expressed in another currency. But this is entirely independent from you transferring coins.