All behaviour is communication.
I would imagine that if his speech is somewhat delayed then he has some frustrations with not being able to communicate with you how he is feeling. That coupled with maybe some anxiety about a new person coming into his space (the room), he is going to the person he feels the most safe with and trying to communicate or rid himself of some of the overwhelming feelings he has.
Play centres are unpredictable and full of rough and tumble and kids screaming and not following any sort of set rules. They are just chaos. He is probably doing the same there.
Nursery is based on schedule and routine. Same staff (most of the time) same kids on set days. Same time for meals. Same time for naps. Same time for outdoor play. Everyone knows where they stand. It's reassuring and comforting. He probably has no need to feel anxiety and frustration when he is surrounded by familiarity.
I would work on practicing the person entering through the door. Maybe have someone come in, so YOU know it's planned. And be ready with a couple of key calming phrases like "it's ok" or "let's see what they are doing together" or something simple at his level of understanding to help him feel calm when it happens and to alleviate any feeling of fear or threat.
If its happening at home, maybe have a rule of people knocking on the door and popping there head through and asking if they can come in first. To give him fair warning. Maybe even ask your child if they can come in so he feels he has some control over the situation.
Having the control at home might just help him manage it more when it is in an uncontrolled environment.
Xxx