Worried job offer will be withdrawn

This is going to be long so bare with me

End of November I started a job at a nursery and I was not taken on at end my probation which was a couple of weeks ago.

During my time I was off sick three occasions, twice was couple of days due to D and V bug (I worked in a nursery) the other was for a back injury that lasted six weeks and I was technically still on sick when I was fired.

I also had to go home early twice with my little boy because he wasn't well. One of these times I said I would take him home and come back as I had someone there to look after him.

Last week I was offered a new job on condition of dbs first aid and two references.

Now what can my previous employer state on reference, including sickness and can the new employer withdraw their offer of employment due to sickness levels?

I'm really panicking as I'm worried my sickness will affect my new job 😞