The rate of tax credits paid are fully dependent on your level of earnings, which means you wouldn’t receive an increase (or decrease) on the statutory rate depending on where in the country you live.
However, you did mention you are also incurring childcare expenses, which presumbably has increased in costs from your recent relocation. This increase may result in a greater payment of Working Tax Credit to meet the childcare costs for your two children, so ensure you let the Tax Credits helpline know that your circumstances have changed as soon as possible, as they will only backdate amounts for 3 months.
Council Tax Benefit may also be available, and we would recommend you check with your new local council, whether you meet certain criteria.
The Department of Work and Pensions produces information on a website www.direct.gov.uk which explains the claim process in more detail.