How to speed up transactions that get stuck?

The good news is that in most cases a transaction that gets stuck because of insufficient gas amount allocated for its execution, can be sped up.

The bad news is that the Safe{Wallet} can't do much about this because the transaction is executed by a signer wallet, so all the speeding up logic is implemented there.

In the wallet UI we merely show a "Speed up" button after a transaction hangs in execution for several seconds.

The same button can be visible in Transactions.

On click one sees an informational message that speeding up should be done in the signer wallet UI.

Speeding up transactions in this context simply means that you resubmit the transaction with a slightly higher gas fee. This provides a bigger incentive for the validators, the transactions with a larger fee go to the front of the queue and are verified first.

Keep in mind that some wallets can't speed up transactions. Another known issue is submitting a transaction with the max base fee significantly lower than the current one. In this case the RPC node won't let the tx through, and it can't be sped up. It will hang in pending state until the gas fee becomes reasonable because of the network conditions change. The only way to get rid of it is to create an arbitrary transaction in the signer wallet with the same nonce.