Your 2-year-old can get free childcare if you live in England and get one of the following benefits:
https://caya-apps.derbyshire.gov.uk/Synergy/SynergyWeb/Parents/default.aspx
At present all 3 and 4 year old children in Derbyshire are entitled to up to 15 hours a week free early education. Many any working families have been able to get up to 30 hours of funded childcare a week.
Eligibility for 30 hours funded childcare
https://www.gov.uk/national-minimum-wage-rates
Families where one parent does not work (or neither parent works) will not usually be eligible. In some circumstances a family may be eligible where both parents are employed but one (or both parents) is temporarily away from the workplace on maternity or paternity leave, adoption leave or statutory sick pay or where one parent is employed and one parent has substantial caring responsibilities or is disabled or incapacitated. These are both based on receipt of specific benefits.
Eligibility will be assessed by Her Majesty's Revenue and Customs (HMRC).
If you meet the eligibility criteria you should be able to take up a place if:
you have a valid eligibility code from HMRC
it is the term after your child's third birthday, or later until the child reaches compulsory school age
you received your code from HMRC prior to the term in which you would like your child to start their 30 hours place
Don't forget even if you are not entitled to the 30 hours extended entitlement your child will still be entitled to the universal 15 hours of free early education.