I doubt very much it would be possible now - maybe 3 or 4 years ago. I was told they could possibly make a case for me to be seen at Tommy's/Kings due to cardiac condition for which I've been treated at Tommy's in the past, but even then it would be close.
Unlikely for MLU Lewisham too - that's what I wanted this time round (but was frankly laughed at due to cardiac issues and prev c/s; MW said even without those obvious no-nos I wouldn't get MLU due to bleeding & PCOS).
Kings is massively under pressure; SIL gave birth at home with ambulance crew having been with her for 3hrs waiting for a bed on labour ward to become available. Other SIL also with kings also gave birth at home as she was told not to go in yet due to lack of beds.
However, I can say that I was really pleased with the care at Lewisham Acute Labour ward with DC2 (3m) - and I got to experience it many times this pregnancy! Had a terrible scan at 12w and the fetal medicine team were brilliant. The high-risk ob I saw due to crap birth with DC1 was considerate, objective and sensible. The duty ob I saw when I was admitted in labour at 24w was fantastic (worth noting that he tried to transfer me to either Kings or Tommy's as it would be safer to transfer baby still in utero - both had NICU places but neither had a bed spare on labour ward). Luckily the anti-ctx drugs worked by next morning, I was in for a few days and cannot fault the care. Admitted 3 more times as emergency before ELCS and always the same great care. Postnatal was also excellent (given it's always going to be a shared ward/shower espec after CS - but MWs were lovely, smiley, always made time, willing to help in any way. Food wasn't great but I was given more when I asked!
Couldn't have been a more different experience from DC1, whom we nearly lost at the hands of QE (no exaggeration - even had to threaten them with legal action just to get the birth notes...).
I'd happily go to lewisham if I ever had dc3. Wouldn't go to QE for anything. And don't overestimate importance of a shorter journey time; I'm not sure I'd have got to Tommy's or kings in time to stop labour. An ambulance will take you to the nearest hospital - so if you needed to call one you'd end up at QE or Lewisham depending on which part of Greenwich you're in.