In the first instance, it's always best to submit a missing mileage claim online:
British Airways - Login
These are usually dealt with in a 2-3 week timeframe.
If the Avios don't show up in your account (and this is often the case where you've upgraded with avios) then you need to call.
Don't call at weekends (fewer staff and supervisors) or early Monday morning (busy with people rebooking for the week ahead). Best time is mid-morning or mid afternoon, when it's quiet (and note it's Easter this weekend, a busy travel period).
I get through to BAEC within seconds and can't recall the last time I waited longer than a minute. No problems with the callcentre whatsoever.
Last year I redeemed 750,000 miles on various flights, all within a day of the date I wanted and in my preferred cabin or one higher.
As far as availability goes, well, it's dynamic so you need to check back regularly. This summer is an odd one as the Olympics are being held in London and this has added a layer of unpredictability and it does seem that BAEC is holding back some award seats; having said that I've booked a reward flight in the peak August period to the US without any trouble using the excellent online availability finder:
https://www.britishairways.com/trave..._gb?eId=106160
Hardly a "second class" experience.
Are you UK based, or elsewhere?