Hey Tasha,
This is my first posting on here but I'll jump in. I doubt you will get a studenship for a part-time PhD, they are usually only offered for full-time PhDs.
Many university website are a minefield of confusing information an are not at all clear about the process for applying for a funded place. Many advertise their scholarships/GTAs and on their website provide an 'apply here' link, which is usually only half of the story and they fail to give clear instructions and links to the right documentation in order to complete the application process correctly.
They usually require that you have already secured an offer before they will consider an application for funding. There is usually no specific time to apply for a PhD place, so you can apply before the next funding round is due. Funding is usually advertised any time from January to May and universities have different application windows within that time.