Bridge layers and the like IIRC were sent to NA -I don't think any gun tanks made it as part of Kingforce (the Churchill armoured unit)
As with all systems in WW2 it changed as it was developed as it was built and the cooling system did have pipes that ran from the front mounted radiator through the crew compartment and the overheating issue while very bad in the earlier versions and while never satisfactorily fixed did get better.
The overheating was noted as making the turret 'uncomfortable' - but none of the notes I have seen suggest that the crews where passing out
If it 'was' sent overseas then I suspect that some of the proposed fixes might have taken place earlier and had more effort expended on it as well as greater effort once in country.
OTL there was enough 'better' tanks that this effort was never required and the Covenanter was 'good enough' as a UK Garrison tank.
I'd far rather have the British build 300 extra Valentines and send them - but I suspect that if they had built 300 extra Valentines that would be 300 extra Valentines on their way to Mother Russia.
Good write up here