What I find bizarre about any supposed 'Anglosphere' if it's to happen is that if we genuinely aim to reduce immigration to the tens of thousands (which has often been the government's target), then any so-called Anglosphere would fly in the face of that.
Under Cameron (and May as Home Sec) we had control of non-EU migration but didn't seek to reduce it - presumably any reduction in immigration would have to come across the board, and thus also in these supposed Anglosphere countries.
I'm all fine for us to have a good relationship with said countries, and I'm happy to celebrate our history and shared culture with them, but I see no reason as to why relations with the US, Canada and Australia should be given priority over major European countries like France or Germany.