i live about equal distance between the two hospitals and have worked at both (not in maternity though). Both are fine. I would say go with the one most convenient to you for antenatal care. I have many friends who have had deliveries at both places, and St thomas's too, and the stories are mixed from awful to fabulous, like with most hospitals i have heard of. I agree with previous poster that post-natal care in most hospitals is not ideal, esp as dc2 you may be able to go home quickly.
I have had 2 births at woolwich, the first a difficult birth with wonderful staff, the second a straightforward birth with a midwife who we just didn't get on with, but all went fine and we went home 2 hours post-delivery. this time i am very much converted and hoping for a home birth. would you consider that if you are scared of hospitals?
i have heard about the lewisham birthing centre which sounds good, i think open in a month or so? you can go and look round both hospitals, ask your midwife/gp or phone up the maternity appointments and they will tell you when the open evenings are.
for breastfeeding, i can highly recommend this group in deptford or look at this list for greenwich area or this for lewisham. perhaps consider going before you have the baby to discuss any issues you had from last time and so be prepared, and also then if you go with baby the group/counsellor isn't a total stranger.
in my experience the post-natal bf support is not great at either hospital, esp lewisham, but i hope this has changed esp if they are opening a MW led birth centre, you would expect them to be better.
there is also a MW birth centre in queen mary's sidcup but i don't know much about that.
good luck with it all and congratulations on your pregnancy.