will / would

He promised his father that he would cycle slowly in future.

This is the "Future in the Past". You have used the past tense to describe what happened during the visit to the park. Therefore the Future in the Past is the appropriate form to use about John's promise.

Incidentally, surely John should have promised his father that he would cycle more carefully in future?

Future in the Past has two different forms in English: "would" and "was going to." Although the two forms are frequently used interchangeably, some U.S. sources state that they can express two different meanings, that "would" is used to volunteer or promise, and "was going to" is used to plan. This distinction is not strictly observed in everyday UK English. That is, in each of these examples, where "was going to" is shown, "would" could equally well be used, and vice versa.

Examples:

* I told you he was going to come to the party. plan

* I knew Julie would make dinner. voluntary action

* Jane said Sam was going to bring his sister with him, but he came alone. plan

* I had a feeling that the vacation was going to be a disaster. prediction

* He promised he would send a postcard from Egypt. promise


Like all future forms, Future in the Past cannot be used in clauses beginning with time expressions such as: when, while, before, after, by the time, as soon as, if, unless, etc. Instead of using Future in the Past, you must use Simple Past.

Examples:

* I already told Mark that when he would arrive, we would go out for dinner. Not Correct

* I already told Mark that when he arrived, we would go out for dinner. Correct