I live in England, I have a bank account which gave me some free shares. I've had these shares for a number of years, the price has changed regularly. I've never sold them or benefited from them financially, but I've begun to consider whether I was required to pay Zakah on them. If I pay Zakah on them, do I pay Zakah for all the missed years since I first received these shares? Is it Haram to accept free shares from bank accounts that deal in interest? Every bank in England deals with interest. Unfortunately there's no alternative to these banks living in Europe. One last thing. I agreed to sell my shares to my Mother at a certain price. Is it allowed for me to sell them to her? Or if it's Haram to sell them to her, do I simply sell my shares to the bank, even though I agreed with my mother to sell my shares to her at a certain price.

Praise be to Allah, the Lord of the World; and blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
Basically having an account in a usury-taking bank is not permissible except if there is no Halal alternate and if one can not keep his money in any way other than depositing it in such banks. This is because those banks use your money to run and increase their Haram activities. Allah Says (interpretation of meaning): {…but do not help one another in sin and transgression. And fear Allâh. Verily, Allâh is Severe in punishment.} [5:2].
It is apparent that the shares allotted to you from the bank are for your cooperation with them and your depositing your money in that bank. Thus, they are actually a form of usury. So, you should not possess them, and should not sell them to your mother or any other person. Instead you should return them back to the bank itself and spend that money in the welfare of Muslims such as paving roads, building hospital, etc. Allah knows best.