It depends. But likely yes you could overlay the existing 19mm solid timber boards with 13mm engineered timber once you prepare them.

you can definitely do it with solid timber boards and nail them through the face in the traditional manner. The floor will be as flat (or not) as the original floor.

With the engineered boards, it depends on the fixing method. If you are doing a floating floor the it’s possible if the current floor is within the specified flatness. If the floor is not flat you will have issues with boards or tongues cracking. You would put an acoustic underlay down first. The floors do creak a bit either as the room heats up/cools down or when people walk on them.

Alternatively you could glue and nail them if the floor under is sound which will give you a far better feel underfoot. The floor board manufacturers literature should be consulted.

To prepare the existing floor boards and improve flatness replace any rotten boards, punch any nails down, flattened with a drum sander and then some levelling compound used in isolated areas prior to overlaying with the engineered flooring.

I would get a flooring contractor in to review and provide advice.

Remember regardless of what you overlay with you may have the same issues with stairs and either all your skirting boards need to come off and be replaced with new (and walls patched and repainted) or a timber bead gets put around the perimeter tacked onto the skirting boards, which is a bit of a giveaway.

Also all doors will need to be remove and trimmed. If they are hollow core doors they may need to have the internal bottom rails removed and new timber rails glued in and likely require the entire door to be repainted. Solid doors can just be trimmed but make sure the bottom edge is repainted.