Two tickets means two contracts. The second airline owes you nothing.
Check your flightOn a single booking, the regulation treats your journey as one thing. A short delay on the first leg that makes you miss a connection and arrive eight hours late is an eight hour delay, and compensation is measured on the total distance from where you started to where you finished.
On two separate tickets, there is no journey. There are two unconnected contracts, and the regulation has nothing to say about the relationship between them.
The first airline owes you compensation only if that flight on its own arrived three hours or more late at its own destination. If it landed ninety minutes late and you missed a connection because of it, that is not a claim under EU261, however much it cost you.
The second airline owes you nothing. From their side you simply did not turn up, and a no-show usually forfeits the ticket. They are not obliged to rebook you, and they will generally charge you a new fare.
This catches out a lot of people who booked a cheap long haul flight and a separate budget hop to reach the airport. The saving on the fare is real, and so is the risk you took to get it.
If you booked everything through one agent or one website and got a single booking reference covering both flights, that may well be one contract of carriage even though two airlines operated it. Look at your confirmation. One reference across both legs is a good sign. Two separate references almost certainly means two contracts.
Some booking sites also sell their own connection guarantee, which is a commercial promise from the agent rather than a legal right, but worth claiming if you paid for it.
If you are deliberately self-connecting to save money, leave at least three hours between the two tickets, and check whether your travel insurance covers a missed connection on separate bookings. Many policies do, and it is usually the only thing standing between you and buying a new ticket at the desk.
Check the first flight on its own