Take these two transactions that happen one after the other:
Transactions:
1) Exchange A -> (1 confirmation) My Wallet
2) My Wallet (With a high fee) -> Exchange B (needs 20 confirmations)
Will the Exchange B wait for the 20 confirmations on transaction #1 as well, or am I able to speed the process up by sending it to my wallet and paying higher fees than the Exchange A does?
When a transaction gets its first confirmation it is included in the blockchain. Barred unusual circumstances, the block that confirms it will become part of the longest valid chain and thus the transaction ever more reliably irreversible.
If you send a transaction to an exchange that waits for 20 confirmations, the previous transaction will have at least 21 confirmations when they accept the deposit. There is nothing you need to do with the first transaction.
