I can only speak from my own experience. I was accepted onto my PhD project in August but due to legal back and forth between the Uni and external company sponsor, I didn't actual start until February of the following year. This was only an issue of no income for those months but there was no issue with the funding, once it had been awarded to the project the Uni had 12 months to start using it or it would have been allocated to a different project and funding would have had to be applied for to the funding awards body.
So a short delay to finish your current degree should not be an issue, as I'm sure they are aware of your studies when you applied for the position.
Good luck!