Hi there
On the missile design screen the improved speed you are seeing is just showing you what you will ge if you design a new missile with the same allocation of space as per your previous design. To get the benefit of the new missile engine tech you will need to create a new research project with those revised stats, research that and then build the new missile. I normally do this by creating a series of missles with the same name and then just add mk2 etc to the end so I can differentiate them.
If you want to look at all your stats for different missles and old tech you can see this from the tech screen summary, can't remember which F button it is off the top of my head but its one of the main buttons at the top of the solar system screen. Click on that then select missles and it will give you a full list of the stats of all of the missiles you have invented.