You will need to register for Self Assessment as a sole trader in the UK. The UK gets the first cut. If your earnings are £43K, you deduct the tax-free allowance, and pay tax on the balance, so you would normally be in the 20% bracket, not the 40%. You will also pay NI contributions in the UK. And yes, you need to get a National Insurance number ASAP.
You do NOT pay self-employment tax in the US. You WILL need to file a tax return, but either take the FEIC to exclude your income or the FTC to use UK tax paid to offset US tax due.
That's the short & simple stuff. Others will come along and provide more specifics.
You need to do a bit of reading