While within one of the EU countries, you could travel by bus and train quite easily, even when crossing borders within the Schengen zone you might not run into trouble. But traveling from the UK to the Schengen zone you will have to show proof of permission to travel from your parents.
I checked out the Eurostar rules, as it is the easiest way to travel from London to the continent. It looks you are out of luck there
Children of 12 years old and above may travel alone provided they have a completed and signed 'Unaccompanied minors form’. This must be signed in front of a member of Eurostar staff.
Airlines are more likely to allow you on board but will need to see permission to travel by your parents and you can not buy your own tickets (if following the rules online, which state you need to be 18 to buy your own tickets.)
There will be some hotels, hostels and 'bed and breakfast' places that do not check your age, but for most of Europe there is the rule you need to be 18 to sign contracts and booking a hotel room and checking in is seen as signing a contract. Hostels seem to sometimes allow 16 and 17 year old teens to check in alone, often a letter of permission of the parents is needed.
Within the UK the same limitations on hotel and hostels exist, with some exceptions where the owner does not bother to check. But at 15 you should not expect to be able to travel long distance without questions asked.
You might be tall, but European kids are often also tall and train and hotel staff know to more ways to guess age. Often hotels and hostels require a passport to check in, and at 15 you will not be allowed and might be send 'home' under police supervision.
If your parents are OK with it and they or your host family can organize it, you might be able to stay with friends of the family somewhere else in the UK, or even on the continent, but you will need to have the right paperwork.