Hello,
We moved to the UK a year ago, and even thoughw e speak English at home, my 6 yr old DS really struggled.
Because the school is small and lovely, he did settle in o.k.-ish. He would sometimes be homesick or sad, and whilst I am always sympathetic, I make a point of downplaying his sadness (ie by not dwelling on it too much, and by highlighting the good things about our new home and school, at the same time making sure I listen to him and give him a hug when sad, if that makes sense).
He definitely sleeps in his own bed, as I have other DS and need my sleep!
Like your DC, my DS is bright and good with maths but struggles with reading and writing. I have had quite a few chats with the teacher, and she has given him some extra homework, and advice.
She has also put him in an "early intervention" scheme, which gives him some extra one on one writing practice with the TA. The next step may be to get him Special Needs help. Having him qualified as SN was a bit of a shock, if I´m honest, but it just means he gets the extra help he needs. The Gvt. has a scheme, it´s cald No child left behind, or something like that, and schools have steps to follow in this kind of situation!
So go ask your teacher about Early Intervention and maybe even SN help. Good luck