Yes, the EU and UK business cycles in economic structures are compatible, because both the EU and the UK have strong laws in place that are positive for business. This is not to say that the UK should join the EU, but if the UK ever were to decide that it wanted to, they would be able to integrate their business systems without much trouble.
Yes, the EU and the UK have compatible business and economic structures because they both have strong economies. The Euro has been solid for some time, and the United Kingdom has a strong economy and a stable business environment. If the UK wanted to, they could join the Euro quite easily with a simple transition. However, that is not to say that the UK should join the Euro, but if they wanted to it would be feasible.