Already in the UK but Visa start date isn't valid yet

I received my UK Youth Mobility Visa on the 19th January 2013 and decided to fly out on the 20th January 2013.

When I arrived at the airport in Australia they told me my Visa didn't actually become valid until 1/02/2013 which was the date we put on the Visa Application as we were advised it could take 3 weeks for it to become back wasn't actually a confirmed date.

The airline said I was unable to travel as it could cause problems for them when I arrived at the UK border.

I then booked a flight to Paris for the 31st of January to show that I would have a return flight and they wouldn't get fined.

on entry into the UK they didn't ask me anything regarding the date or what my intensions were however on the arrival card I did say I was leaving on the 31st to Paris.

My Visa for the UK was stamped and now I am just visiting a friend before I start working in the UK.

My questions is, will I still have to leave the country on the 31st as I am already in the country or do you think I will be right to stay in the country until my visa comes into the vaild date on the 1/02/2013?

Any advice would be appreciated!