When BTC is mined, it is sent to the BTC wallet. The duration of the transaction can be up to 24h for online wallets. Is the duration faster, when transferring BTC to a “cold wallet”?
It's helpful to keep in mind that a wallet, is just an application that interfaces with Bitcoin by managing your keys associated with your bitcoin. So if you broadcast a transaction, the transfer time you're talking about, is actually the time it takes for the miners to include your transaction in a block included on the blockchain. The time it takes doesn't rely on the wallet type you're using, but the actual Bitcoin network.
If you are waiting 24hrs, it could have to do with a low transaction fee. If the mempool is full, the miners will generally choose transactions with higher fees (to maximize their profit). If the fee that was paid on your transaction is relatively low you'll have to wait for the mempool to empty out before the miners include it in a block.
If you're relying on a third-party, they may be doing something else that isn't transparent from the outside, and is causing a longer wait time.
Coins are spendable by a wallet as soon as the transaction is included in a mined block. Some online wallets use their own software to sync, or in other words, to make their application aware that the account has received coins. Applications that use this method could have varying durations for recognizing received coins. Furthermore, different wallet providers may also have additional criteria, such as requiring that at least 4 blocks have been mined after the transaction was included in a block. This is known as Minimum Confirmations. For example, Coinbase will credit your account once there have been 3 network confirmations. In this case, the duration to "receive" coins on a local wallet such as Bitcoin Core, would be much faster than on Coinbase, because you would not have to wait for 3 blocks in order to spend the coins. It's also worth noting that this is not an issue of "online" versus "cold" wallets, it's more of an issue of third-party service based wallets versus personal wallet clients which you control the keys for.
As @JBaczuk pointed out below in the comments, technically coins are spendable even without a confirmation. This is especially useful for child-pays-for-parent (CPFP) transactions, where the first transaction may have a fee that is too low, but it is used as an input in a second transaction with an extra high fee, so miners will include the second transaction, which would also require including the first transaction.