As it stands, the letter you've written has no validity. When you die everything will go to your mother. If she predeceases you then it will go to your aunts and uncles (if any) or otherwise to the Crown:
http://www.hughjames.com/wp-content/uploads/2014/03/Intestacy-Rules-October-2014.pdf
(There would, of course, be nothing to prevent your mother - if she was still living - from honouring the terms of your letter by voluntarily giving half of what she received to your friend).
However you don't have to "complete a formal will and have it countersigned blah blah by officialdom". OK, you do do need to draw up a proper will (see below) but there's no need for any 'officialdom'. My own will was witnessed, by a couple of people I simply happened to be working with that day, in a Tesco car park!
Copy and paste the following onto a blank sheet of paper, changing the names, addresses and other details to meet your requirements:
"This is the Last Will and Testament of me, James Bond, of 42 Acacia Avenue, Newtown, in the county of Nowhereshire.
I hereby revoke all former Wills and testamentary dispositions made by me.
I appoint Fred Bloggs, of 46 Acacia Avenue, Newtown, in the county of Nowhereshire, and Martha Bond, of The Larches, South Road, Anothertown in the county of Whatshire as the Executors and Trustees of this Will. Should either of the aforesaid persons predecease me or be unwilling or unable to act as Executor and Trustee I appoint the other person alone as sole Executor and Trustee.
I desire that my body be cremated after a ceremony in accordance with the traditions of the Church of England.
I devise and bequeath all my real and personal estate whatsoever and wheresoever situate unto my Trustees, upon trust, to sell, call in and convert the same into money, with power to postpone such sale, calling in and conversion for so long as they, in their absolute discretion, shall think fit, without being liable for loss.
My Trustees shall hold the net proceeds of sale and conversion of my ready money, upon trust, to pay thereout all my just debts, funeral and testamentary expenses.
Subject as aforesaid, my Trustees shall stand possessed of the residue of my estate, upon trust, to distributes in accordance with the following provisions:
Fifty percent thereof to my mother, Martha Bond of The Larches, South Road, Anothertown in the county of Whatshire.
Fifty percent thereof to my best friend, Fred Bloggs, of 46 Acacia Avenue, Newtown, in the county of Nowhereshire.
Should either Martha Bond or Fred Bloggs predecease me, I leave the whole of the residue of my estate to the surviving beneficiary.
In witness whereof I, the said James Bond, have to this my Last Will and Testament set my hand this tenth day of September in the year two thousand and fifteen.
[Space for your signature]
Signed by the said James Bond, the Testator, as and for his Last Will and Testament in the presence of us both at the same time who, at his request and in his presence and in the presence of each other, have hereunto subscribed our names as witnesses:
[Space for the signatures of the witnesses] "
Then all you need to do is to find two witnesses (who must NOT be your best friend or mother), sign your will IN THEIR PRESENCE and get them to add their signatures underneath. (It's best if they print their names and addresses as well).
After that, all you have to do is to store your will in a safe place and TELL YOUR MOTHER AND BEST FRIEND WHERE IT IS!!!