Laying hardwood floor over sagging original floor

If the floor is sagging I’d say it is important to lift the existing flooring and address the reason for the sagging - this is likely to be the joists that need sorting. Once the joists are sound you could buy pine floorboards that are inexpensive and would provide a more structurally sound wood floor when their tongues and grooves are locked together.

If there a sag to the floor it would suggest the joist have bowed or broken and would be in need of replacement. It could be covered under buildings insurance as a structural part of the building.

I would say get some advice regarding the tiles you have down now before doing anything else as they could very well contain asbestos especially the black adhesive which is more than likely asphalt tile adhesive and definitely contains asbestos.

The sag in the floor needs addressing before you spend any money on floor boards. Some or all of the exiting boarding over the joists need lifting to find out why it's sagging. Remedies as simple as putting a row of strutting solid or herring bone (see Google)down the center of the floor between each joist will help take out any bounce and stop it sagging Futher, or bolting timber (4x2 or 6x2 etc) depending on the size of your joists to the the upper side of each joist to level it out hopefully above any services running under the floor. Then you can put what you like down, hardwood or softwood tonge and groove . Don't board over the problem!