End of Marternity Leave - Starting a new job during notice period?

[FebruaryBear]

I am currently on Maternity Leave - my SMP has just finished but I am entitled to be on leave until January.

For a few reasons I have been looking into other options, and have now been offered a job which starts in 2 weeks time, which I had wrongly thought was also my notice period for my previous job. In fact, my notice period is one month.
My contact states I have to ask for permission to take a second job if it would affect my ability to do my current job - which it won't, as they aren't expecting me there anyway (or paying me).

What would you recommend? Do I need to tell my old job my new job starting date? What would happen if I didn't?
I only took SMP so don't believe I'd need to repay anything, and I do have around 3 weeks of holiday accrued.