Thoughts on the slow start of the above Car?? See videos.... I'm thinking battery/alternator? Anyone experienced this?

2011 passat cc 2.0 TDI Bluemotion slow start - YouTube - From cold

2011 Passat cc 2.0 TDI Bluemotion slow start WARM - YouTube - From warm

Makes no difference, maybe a smidge faster when lights, heater and radio are off for the start.

Dealers problem as only had car since Thursday and under warranty, only driven 100 or so miles. Hence why suspect maybe battery.

Car is full service from VW, 67000 AND JUST HAD TIMING BELT AND KIT fitted with a service and MOT.

Any help appreciated, hope the discussion helps a few others too.