Can I just say that it's not worth following the trend against checked baggage and causing yourself inconvenience? It's so not worth it.

A one-week trip means I cannot take my favourite body wash, shampoo and conditioner as they won't fit into a 1-litre liquid bag along with other liquid based cosmetics. I will also incur laundry cost at the hotel because 1 week's change of clothes will not fit into a small carry-on bag along with everything else. You might not be able to fit an alternative pair of shoes or flip flops into the bag. Overall, it seems to me to be a way of causing stress if you are pushing yourself too much to go with hand luggage only. Admittedly it's easier because it's summer but if you are dreading packing for it, it's not worth the stress.

How about starting on a shorter trip and then extending the period you can do with hand luggage only?

Aside from that, there is an exceptionally light 4-wheel cabin sized bag from Samsonite called 72H. It's not suitable for frequent user (perhaps more than 2 flights per month on average) as the wheel strength and stability are not great, but it is extremely light, so it may help you to be able to put the bag into the overhead locker yourself (which is a requirement on BA).

P.S. Unless you are pretty strong, I do not recommend a wheel-less cabin bag due to stiff shoulders you may incur while walking around the airport, and you may not be able to run as well if you are running late.