In the UK, cash paid at the institution in question clears the same day. If bundled with a cheque (rather than as two separate transactions) it will clear at the same time as the cheque.
I had something similar happen to me many years ago. I took ATM cash from another account, plus what I had in my house and wallet, to make up the amount that an auto-paid credit card bill due on that same day would have sent me overdrawn.
The most absurd thing was that the bank, RBOS, callee me few days after about the late fee asking me if i was having money problems... It was a bill under £2 that had just slipped my mind. If they had checked any of my other accounts they would have seen that i had about £50k with them.. . But no a £2 bill has them calling me almost sounding like they were shaming me over an inconsequential amount of money!
I had something similar happen to me many years ago. I took ATM cash from another account, plus what I had in my house and wallet, to make up the amount that an auto-paid credit card bill due on that same day would have sent me overdrawn.
I avoided the overdraft penalties.