You have the right to take reasonable time off to deal with an emergency involving your child if he/she is ill and you need time to make arrangements for their care. The employer does not have to give you paid time off, but many employers do give paid leave in these circumstances, e.g. up to a certain number of days a year.
If your employer is refusing you time off for this purpose you can refer the matter to an employment tribunal, but there are strict time limits for making a claim.
There is a helpful free leaflet from the TUC, called Time off for families.
The charity Working Families also has useful resources and information.