We took our small boy long haul (London - FLorida on BA) when he was only nine months old. He also did short haul (London - Hamburg) when he was only three months with no problem. For the long haul I would suggest that you book a proper airline (as you have it seems) and get a bulkhead seat with cot (which you hopefully have) and all will be fine. The only thing I would disagree with everyone else is taking a push-chair. Apart from the fact that as someone with a professional as well as personal interest in cabin safety, I seethe massively at excessive cabin baggage, as a parent I would say that far from being necessary it will be a great inconvenience, especially on your own. Even at nine months, I carried him in a soft (i.e. non-frame) sling thus avoiding the inconvenience to myself and everyone else of a push-chair. We bought a cheap second hand push-chair on arrival in the US then returned it to a charity shop as we left. Give that a go maybe?
Amusing but serious, I have never been forgiven for saying to my other half as we strapped in "If we have an emergency evacuation, don't look for me or Robin, just go". She interpreted that as me saying that I was going to trample her if she got in my way with Robin! Travelling alone, you won't have that problem!