The Registrar of Companies is the quick part. Aqualis sees name approval in one to two working days, and an accelerated incorporation is usually through in about a week. If incorporation were all you needed, you could be trading in a fortnight.
You cannot, because customers need somewhere to pay. A bank will not start its checks until the company exists and its certificates are issued. Then it verifies every shareholder and director, the source of the money, the expected flows and the main customers and suppliers. For a single founder with a clean file that takes three to four weeks. Where a shareholder is itself a company, the bank repeats the exercise up the chain, and two months is normal.
Registrar time is about one quarter of the programme. Bank time is about three fifths. Spend your effort where the weeks are.
The practical answer is to prepare the bank file while the Registrar is working: certified passports, proof of address, a short business plan, contracts or letters of intent, and evidence of where the funds come from. Some founders open an account with a licensed electronic money institution as a bridge. That can work for invoicing, but check that your customers and your auditors accept it.