Welcome to flyertalk citiflyerUK!
I have recently done a few upgrades that were part of BA holiday bookings and yes it can be done.
First of all with regards to doing it at check in (and I am sure others can verify) my experience is you can only pay to upgrade to the next class and even then its down to what's available so you take your chances and will only be able to get a WT+ seat if at all.
I called up recently to upgrade the outbound leg of my BA holiday booking from WT to WT+ and was given a price (about £700 each I think) that I wasn't happy with so didn't bother. A few days later however I called again and managed to get the upgrade for about £350 for both of us so its worth calling and seeing what they have (they were also able to upgrade me to CW but the price was too high for me). I did notice that the price difference was what was showing online when I did another holiday booking and there was no charge for a change, although some on here have been charged a change fee in addition.
If you have some avios its also worth upgrading to WT+ and then doing a UUA if there is availability as this will likely be a lot cheaper than outright J fare and that's what I did with the above booking. If you don't have the avios (and want CW) it may also be a lot cheaper to buy the points and still do the UUA rather than paying the fare difference.