You can booked timed tickets at https://www.barbican.org.uk/whats-on/2025/event/visit-the-co... but when it's open on weekends, one can also often just walk in. It's a great place to visit. Entrance is free.

As far as I'm aware, the Barbican Conservatory (Greenhouse) will close for refurbishment at a point next year though. When you go currently, they'll have details of the plans for public consultation. So see it while you can (or then again in 2030 or so).