Do we need to preheat both the fuel e.g. LH2, and the oxidizer e.g. LOX, by running conduits around the nozzle? I presume both being cryogenic, preheating would be a must otherwise, engines will have low efficiency
Actually there is no need to preheat the oxidizer nor the fuel.
Thermodynamically it even lower the efficiency.
If there is liquid running conduits around the nozzle it is there to cool the nozzle!
You can improve the efficency by getting the exhaust gases hotter, but to hot would melt the nozzle.
So you are "investing" a lost by heating your fuel/oxidizer for a higher temperature at the output. That is "giving you more efficency" than you have "invested" in the first place.
You are basially playing with Carnot's theorem.
So you use a part of your cold fuel to cool the nozzle.
an example: Wikipedia Rocketdyne F-1
some of the fuel first traveled in 178 tubes down the length of the thrust chamber — which formed approximately the upper half of the exhaust nozzle — and back in order to cool the nozzle.