No! You cannot work in the UK without the national insurance number. You can start working if you are able to prove that you have the right to work in UK. But later you have to apply for the number.
You can apply for the national insurance number if you meet the following conditions:
- You are living in the UK
- You have received the work visa and get the right to work in UK
- You are seeking a job, or working, or have an offer letter to start work
There is no need of a national insurance number if you are starting an application for the benefits or student loan. If you want to apply to the European Union Settlement Scheme, the national insurance number is not needed.