I am a Japanese national working freelance for a Japanese company and for personal reasons I will move permanently to the UK on November 13th this year. The Japanese company will let me do freelance work for them even after I move to the UK. I plan to work for them only until I find a stable fully UK-based job. For this, I will have a monthly income which will be over the £2000 allowance, but I do not plan to transfer it to my UK bank account. I read the articles about the foreign income taxation but I am still not sure what I should do. I was told by my current company that I will not pay the taxes in Japan.
Assuming you will just freelance as a self-employed "sole trader" and don't intend to set yourself up as a limited company:
That'll probably include registering for self-assessment with HMRC.
It should not matter that the income is from foreign sources and/or paid into/retained in a foreign bank. So far as HMRC is concerned, for the purposes of assessing tax liability "you would just convert this income into pounds sterling and add this to your turnover." (source); however as that notes: if tax is deducted in the foreign payer's country you might also need to fill out the "foreign section" of the tax return to avoid double taxation.