diagnosing a3 1.6tdi issue

Hi - I have a 2013 1.6 tdi audi a3. It's done 163k miles, running beautifully and still getting great economy! My issue is that it takes a while to start from cold. There seems to be plenty of life in the original battery still (yes - has never been changed!) but wonder if the glow plugs are at fault.
I don't get any rough idle, or the glow plug icon flashing on the dash. I do get the yellow check engine light on ignition, which then goes off when it starts. It just takes about 3 or 4 seconds of cranking to start when cold, and then when warm starts almost instantly. If I turn the ignition on and off a few times from cold to repeat the glow plug cycle a couple of times, the symptoms ease slightly.
With winter round the corner, I'm thinking I should not let this go. Do you think the glow plugs would only be an issue IF the glow plug light comes on - which it doesn't, so this is more the battery than the glow plugs??
Any help / advice would be appreciated.
Thanks
Julian

The CAYC has active glow plug monitoring and they are used as part of the emissions strategy during normal running so in theory, a failed glow plug should cause the emissions warning light to come on when running. In my experience, it is more likely that the cranking rpm is below the critical value of 280rpm, usually down to a worn starter motor and very common with the Valeo brand. The standard test is to unplug the injectors and crank the engine over whilst observing the rpm in measuring blocks using a suitable diagnostic system. After reconnecting the injectors, the fault codes generated doing this test need to be cleared.