You should be fine and, you're correct, you should carry a document from your parents.
Documents you need for travel in Europe
Documents for minors
In addition to their own valid travel document (passport or ID card), although not obligatory by law, all minors entering or leaving Poland are strongly advised to carry a letter of parental consent if travelling alone or with adults other than their parents. The document should be signed by both parents and certified by a notary (or validated by a Polish Consul if issued outside Poland) and should show:
- the dates and reason for travel
- the details of an adult who will be responsible for the child
Find further information in the Guide issued by the Polish Border Guard.
As for your entry into the UK and your departure:
Other than their own valid travel document (passport or ID card), no particular extra official authorisation is required for minors of any EU country to enter or leave the United Kingdom. However, if a child who is resident in the UK is travelling with an adult who does not have parental responsibility, permission will have to be obtained before leaving the country.