I'd like to see a reference for this, it's possible the second language has an influence but children entering education in the UK are normally fluent within 6 months. They will not have the same vocabulary as an adult because children don't have the same vocabulary as adults in any language.
I said "100% up to speed" not "fluent."
There is "superficial fluency in everyday language" and then there is "being able to manage more difficult skills at the same level of everyone else." I gave the example of describing a book that one is reading as an example of the kind of task I am thinking of. It takes a couple of years (or more) for kids to be fully up to speed with these higher-level tasks. I'm going on what I've been told by people who work with children like this in actual school systems.
They will not have the same vocabulary as an adult because children don't have the same vocabulary as adults in any language.
It's not about having to have the same level of vocabulary as an adult. It's about having the same level of vocabulary as other children their own age. It takes years to build up a strong vocabulary.
Children's ability to learn new languages before puberty is impressive, but please do not underestimate how much time and exposure is acquired, and please remember that "surface fluency" in everyday conversation is not the same as being able to participate in a lively classroom discussion, follow complicated instructions or give a verbal report of something.
In Japan, I'm constantly seeing a string of people who turn up in Tokyo on a whim, and plonk their 9yo (or whatever) in Japanese public school, eagerly chirping that "Children are magic sponges! They learn languages in a few months!!" Then before long they are howling for help as their child starts to struggle badly.
Admittedly, Japanese presents particular challenges because of the difficulty of the writing system. But I've also seen similar issues when kids with a poor level of English are put into international schools. Yes, they can get up to speed, but it takes time. You need to think about how to support them and fill in gaps, especially in terms of vocab and writing skills, and be patient in the meantime. Especially if they are, say, in the later grades of elementary school when they are switched.