You can totally start now. Although, and maybe this is just a me thing, I’d feel like a massive bellend if I referred to something that happened in, for instance, 2021 as “in '21”.

I think I’d feel okay with “'01”, through to “'09”, then the teens feel weird again but only because it just feels weird to refer to a year as small as like… “'13”… although I don’t have that same problem with the naughties, maybe that’s because of the added “oh” making it seem like more than just a number? And then the twenties feel like big enough numbers to abbreviate but, yeah, again, I’d feel like a tool.