You don't need a permit (England & Wales).
Scots families do need permission from their LEA, and so do all UK families using a Special School This doesn't mean statemented children. For either Scots families, or Special School users, permission must not be witheld from you unreasonably.
For home education support around Special Needs contact Home Education Special

In England and Wales you are not required to notify any authority,
except if your child is in school already, when you MUST notify the school IN WRITING.
You are required to write to the school to tell them the date your child is leaving their school, exactly as you would if you were changing schools. This is so the school do not continue to draw public money on behalf of your child as their pupil.
You may prefer to simply state in your letter to the school that your child is leaving, and that you are making arranging private education. Some schools still don't like homebased education and can sometimes try to argue with you in an inappropriate way. Occasionally they do not even know the law themselves and will try to say you need their permission! This is an unnecessary annoyance for you at such a time so you may wish to avoid it by stating the minimum information required, in your letter.
You may need to prove later you did write to the school, so send the letter by recorded delivery and do keep the post office chit safely. Alternatively, deliver the letter by hand, but make sure you get a signed receipt for it from the school office.
If your child completes primary school education, you do not have to notify anyone as they are automatically deregistered from the primary school.

The school then notifies the LEA.
The LEA normally contacts you to ask you for information about your plans.
Even if your child has never attended school you may still be contacted by the LEA so please read on.

Your relationship with the LEA is important as they can order you to put your child back in a school which may be, in your eyes, unsuitable.
Keep all LEA correspondence IN WRITING. If an LEA officer, teacher, Social Worker or education welfare officer contacts you it is best to say politely that you prefer all correspondence to be put in writing so you have an accurate record. Do insist on this, as friendly telephone chats with authorities all too often end in tears! because what is said is not remembered the same way by both persons.

You are entitled to a "settling in" period of up to 6 months. This is not law, but it is considered a "reasonable period" - and "reasonable" is the key issue in law. You may well feel you need time to research and explore what exactly you are going to choose to do, so just reply in writing to the LEA and tell them so.
ALWAYS ANSWER ANY LETTER FROM THE LEA WITHIN TWO WEEKS. This is so you do not give the impression you are avoiding them, which makes you seem to be a "cause for concern" that needs investigating. Always reply, even if it is only to say that you received their letter, but you haven't had time to prepare a full reply yet. In this case, say when you can write to them more fully e.g. another week, or another two weeks.
If you are requesting a period for settiling in you should state a date you will be willing to supply them with full information. Then be careful to note the date as a family deadline and contact the LEA at that time. Otherwise you will be perceived as irresponsible and a "cause for concern", which can lead to a referral to Social Services under Child Protection guidelines, or to a School Attendance Order forcing you to put your child back in school.

Your LEA might send you a letter ordering a meeting with you and requiring you to fillout forms. This is not correct. You are not legally required to admit LEA officers to your home to 'inspect' you, nor need you fill in LEA forms sent out to you to describe your educational plans.
It is very much your choice, in law, HOW you give the LEA information about your educational provision.
You can supply the LEA with your own written report about your child's education (which any of the HE organisations will help you prepare).
Or you can choose to have a meeting. Some parents feel a meeting is quicker and easier, others prefer keeping things in writing. If you are unsure contact the organisations listed below.
You can arrange for a meeting in a neutral place: neither the LEA offices nor your home (e.g. the library). Your child/ren does/do not have to be there and another HE parent can be with you for support and to witness what is said. Many people find it useful to tape any meeting. It is best to be this careful as this meeting is written up as a formal report which can affect future arrangements. Always ask for a copy of the report.

The LEA only has a duty to investigate you formally if they receive specific evidence (e.g. from a Health Visitor) that your child appears to be seriously lacking in educational provision. In this case you must provide a clear, reasonable account in writing of the education of your child, or let an LEA officer meet you and your child. If your relationship with your LEA is not going wll seek help from Education Otherwise, Home Education UK, or First College UK, see links below.