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Thanks for your question -I am Sam and I am one of the UK tax experts here on Just Answer.
You do not need to advsie HMRC of your current status - as they will see you are either
1) Employed/self employed
2) Unemployed and claiming benefits
3) Unemployed and not claiming benefits
4) Claiming another sort of benefit
5) In receipt of pension
As they have no source of income for you - they will know you are unemployed and no benefits - you only consideration will be whether your income and severance in total saw you in excess of the 20% rate band ( threshold of income more than £42785) If so then you will need to complete a self assessment tax return for the year just gone - so then do alert HMRC (Via their website) that you wish to register for self assessment and it may be that you are due a refund if you did not use your full quota of personal allowances for March (although with the severance payment I suspect you did )
Please let me know if you wish me to expand on any of the answer provided.