Is it okay to work on a research project with a professor at a different university, in addition to main PhD project?
In the UK the funding system is somewhat different to many other countries: PhD students are either self-funded, or receive stipends through Centers of Doctoral Training or from University funds. Since you are not funded from you professor's grant, you do not work for him/her. You work on your PhD project to become an independent researcher, and as such, you can (and sometimes expected to) make your own choice in how the project is executed.

Of course, a simple courtesy requires you to ask your supervisor first before collaborating with someone else. One obvious reason is that you probably don't fully know the relations between your professor and other colleagues and could potentially disclose some of your supervisor's research ideas to a colleague who is more of a competitor than a collaborator. Academia is a twisted world sometimes and it is although you have full right to manage your own PhD project, a bit of information and advice would never hurt.