Maternity rights with fixed term contract

Hi. My wife is in her early stages of pregnancy, and we're a bit unsure how we stand regarding maternity pay. She's worked at the same company for 5 years, but it's been 1-2 year contracts which have been extended each time it's come to an end. This is as a PAYE employee, not a contractor.

Her current term ends mid-December this year, and whilst it probably would have been extended, she wouldn't know about it until nearer the time anyway. When her current term comes to an end, she will be in her 38th week of pregnancy. Is she entitled to anything after her term finishes if they don't extend her contract? Or does the end of the contract end any entitlements that she has from her work?

If not, is she entitled to maternity from the government? On the goverment site, it sounds like after her contract finishes, she could just start claiming maternity allowance. However, in a PDF on the workingfamilies .org. uk website, it states under week 29:

(Sorry, MSE isn't letting me post links as this is my first post)

And we wouldn't be able to claim on week 29, as she'd still be working. Does this mean we can't claim this at all then?

We're hoping that her work will extended her contract as they have done the past few years, so this shouldn't be necessary. We'd just like to know where we stand if it does end.

Not sure if it makes a difference, but I'm also working full-time as a PAYE employee.

Thank you for your advice,
Dan