No, it is not Riot's problem that you forget how to access an account you owned and thus cannot change it's username. That is your problem - you forgot it. Riot is more than happy to assist you in recovering the account(s) if you contact the respective region's support staff for help. Aside from that, they are not going to force those name changes on those accounts - because you cannot prove you own them unless you... prove you own them.
You can ignore the popups all you want. In January, you will no longer be able to play League (or to access friends, the shop, server transfers, etc) until you change it. In February, you will not be able to log in to League of Legends at all. Period. This is not changing. It is not Riot's problem that you forgot login credentials to accounts you own.
No company will ever simply give you something you have no way of proving you own - in this case, you are asking them to force-change other accounts' usernames when you do not know the password to those accounts and you refuse to go through account recovery processes. Imagine if I could go to your bank, say "hey, yeah, this is totally my bank account... I just don't have any proof it's mine or that I'm the person listed on it... but trust me!!!!!!!!"
You further have absolutely no way to know that you are the only one with that login name - there is absolutely no way for you to know this, and as such, your suggestion of a timeframe is irrelevant. Riot also considered this as well as "first in" solutions - where the first created account would retain the name and all further accounts on other regions would have to change. Both were considered to help so few people and cause potential harm to others (i.e. maybe someone created an account on EUW 3 months before moving, then created their NA account, and wouldn't be able to keep the NA account's name because their EUW was technically created first) that they were not workable solutions.
