Notice Period Confusion

I handed in my notice. Took a week before anyone replied.

Then HR had me in a meeting in which they gently suggested I try to get my new place to have me start ASAP. Because they don't want to put me on a new project if I'm leaving but they also don't to pay me for nothing.

So I asked my new employer and they moved my start date earlier.

I informed HR that the date was changed and I could leave as early as next week.

They replied saying they'd prefer to have a chat first and that I shouldnt be to hasty to leave. We are now two weeks in from my notice and I still don't have an agreed upon exit date. And this "meeting" is mid next week!

What's more is that I can't keep going to my new employer and changing dates because that's gonna look like I'm unorganized.

How can I handle this professionally? Because I'm really starting to feel messed about.

Btw I'm in the UK. My contract asks for one month notice.