Welcome to FT and the BA forum
oldnewbie. Sorry that this unfortunate situation has brought you here but hopefully you'll pick up some advice to make the most of this trip and learn some tricks and tips for the future.
As you've learnt, kids under 2 are classed as infants and don't get a seat of their own, unless you choose to pay a child fare. BA do have bassinets (cots on a shelf
) at the first row of each economy cabin which make life easier. Depending on the size of baby, you can bring your own car seat if it meet requirements.
Reserve your seat for free
When you travel with an infant (under 2 years) who won't be travelling in a seat of their own, you can reserve a seat for yourself and everyone in your booking, free of charge, as soon as you've made your booking. If you don't choose a seat in advance, we'll allocate a seat for you three days before departure.
If your infant is less than two years old and you haven't reserved a carrycot position, your infant can sit on your lap using an extension seat belt. The cabin crew will provide this for you and show you how to use the seat belt.
Also, you can choose seats at the time of booking to ensure you get a bassinet ... so sort this out right away before anyone else beats you to it!
http://www.britishairways.com/en-gb/...g-ready-to-fly
Regarding upgrading to WT+, you've also learnt that you can only upgrade using your Avios from the highest economy fares (Y, B and H). Your best bet is to keep an eye on MMB (Manage My Booking on ba.com) to see if any attractive cash offers appear. These can be much less than the fare difference you've seen, but depends on how well the flight is selling.
You'll also see reference to cash upgrades at the airport which can also be very attractive, but at that late time you're unlikely to secure a bassinet seat at the bulkhead unless loads are very light.
Hope that makes sense.