However, the continuous flow of cold fuel would grossly overcool the oil unless the bypass diverted oil around the FOHE to maintain the bulk oil temperature.
All the fuel flow always goes through the FOHE, but at low fuel temperatures, most of the oil goes around the FOHE via the bypass.
Sooty, I see you have worked for RR so I must bow to your knowledge. I only know what I was taught on the AMEL course, and read in the AMM (which is a bit basic on the B777)
The last iteration of RB211 the -524G/H on the B744 and B763 had a very complicated system on two FCOCs with bypass valves for low fuel temp and high fuel temp. But these only controlled the flow of oil, not the flow of fuel.
But engines transfer large quantities of heat to the oil, which is passed to the fuel.
The Trent 800 also has an AOHE, air cooled oil cooler. This is controlled by the FADEC to cool the oil when the FOHE can't cope. If the AOHE control system fails, the valve is driven full open. It is covered in the MEL and can be locked part open. There is a caution in the MM
"Make sure you lock the AOHE in the open position. If the AOHE is not locked correctly, oil and fuel temp can rise much higher than normal and cause damage to the engine"
Do you have any reference to how the FOHE Oil bypass valve works? The AMM just says low temp.
The operating oil temps are 50-185 degC, and norm about 100-120. What is low?