Engine won't turn over - Extreme cold?

I have a 1.4 l petrol 2003 Ford Fiesta Zetec with only 58000 on the clock. It had been working perfectly fine until Monday when suddenly it didn't! We have had 2 bouts of extreme cold wetaher where I live (was -19.6 deg C) but this didn't seem to affect it at first. Anyway, the engine wouldn't turn over, the ignition would make all the normal dash lights appear and switch off as normal, the headlights all seemed normal so didn't think it was the battery. We tried jump starting anyway and that did nothing, so we bump started it which worked immediately. The engine was then switched off and I could get the engine to turn over straight away. Which makes me think it can't possibly be the starter. The alternator is obviously charging the battery as the lights still worked. I took it to a mechanic who thought as it was the car's original battery, should replace the battery as the volt meter was kind of in between a normal reading on a test, but this has not resolved the problem. I have put blankets on the engine whilst still warm tonight, hoping that this will let the engine turn over in the morning without a need to bump start. However, there is clearly an underlying issue here and I'm wondering what it probably is and if anyone can help me out! Thanks