I'm not British, but I figured it's several hours later where you live than in Great Britain, and you might need this before the British wake up.  So here are my thoughts on what you wrote.

Corrections in red.  Stylistics in blue.

British English speakers' help needed: could you please check the phrases for a game for children?

Dear all, I've translated our Russian scenario of playing with patterns into English, but I'm not sure it all sounds genuine like British English, so could you please be very critical and correct it where it doesn't sound right?

Here I go:

Let’s learn about patterns.
Blue red blue red blue … (I place buttons in front of a child)
What comes next?
Blue
Not exactly.
Look, blue and red repeat over and over again. Look at the order of the buttons.  [1]  It looks beautiful. Blue red blue red.  But if we put red again here, it’s not going to look beautiful. 
Let’s play again. 
Red green red green red
Can you tell me which color will be next?
Green.
Nice job!
Let’s try a different pattern.
Cat dog cat dog cat dog
What animal is going to be next?
Cat!

Fabulous.

[1]  "chain" is not correct.  The formal term is "sequence," but I doubt many children would know the word.  So I expressed the same thought in a way a child would be likely to understand.