"I'm not saying that people should give you any special treatment because you've got a baby and young child but a little consideration would have been nice."
So, you are then. Not a problem, but don't pretend you weren't expecting special treatment.
"On the dlr, nobody offered their seat even though i was holding a baby and very obviously wobbling all over the place."
Most people are listening to music or playing on their phone. You could have asked someone in one of the seats near the doors if they wouldn't mind letting you sit down. I've never not seen someone jump up when actually asked.
You had kids, so your reason was obvious, but anyone who has a less obvious reason, you can request special cards from TfL to show to demonstrate you need a seat more than other people.
"One of the lifts was broken at a station so dh carried the pram down some steps and I took the children down the escalator."
That's unfortunate. Sometimes when stations temporarily stop being step-free, you don't get enough notice to re-plan your journey properly. What I would say is that if your DH had needed help, or if you'd been on your own, someone would have helped you carry the pram. That's the one thing we're very good at in London - helping people with prams or suitcases without being asked.
I've helped carry many a pram in my day.
"I had dd in my arms and ds holding my hand."
The way you describe it, that's not safe at all. You need to be holding the handrail. If you couldn't get down the escalator without holding the handrail, you should have asked station staff for help (if there were any) or made an inconvenient detour to a step-free station and transferred to the bus.
I'm not saying this to be mean; I'm saying this because so many people have accidents on that thing each year. In pretty much all cases, it's because they've been travelling without gripping onto the side. It would have been horrific if you and/or DS had fallen. Please don't be tempted to do that again!
In London, you stand on the right, gripping the handrail, so people can move at high speed on the left without knocking you over. It's not a stand in the middle hands-free job. Even if someone had been walking slowly down the left, hugging the left, I can see how that could have made DS unstable if holding onto you and nothing else, with you holding onto DD.
"A man pushed past to overtake us and walk down the escalator and nearly made ds topple over. Ok, I get people do that on busy working days, but on a Saturday afternoon?!"
People are always busy in London, all of the time. Saturday afternoon is busy for many different reasons.
"And when we did manage to get into another lift at a different station, 3 men got out and said to us to give the lift a few minutes airing time as they'd been smoking substances in there."
Sounds pretty convenient for a bunch of druggies, TBH! No one ever gives me notice! Just being unhelpfully blasé about it - I can understand why this freaked you out.
"There were many moments similar to this during our day out. Maybe we were just unlucky but it really put me off going on public transport in London with young children. It has left me with the impression that it's just every man/woman for him/her self. We're from SE Essex so not exactly a sheltered, rural place. AIBU over this or can anyone relate?"
To an extent, yes. Public transport in London is absolutely great unless you need step-free access, then it's suddenly shitty. It's the absolute worst for wheelchair users though - I've seen many not get onto a bus because parents have refused to collapse their prams. And I've heard many stories of wheelchair users turning up at a station and - just like in your experience - suddenly finding the advertised lift is out of order, so they can't get off the platform. I freely admit London transport is only excellent if you have no mobility needs, and that's not an acceptable standard to have.
Holding onto two kids yourself is what made your travel so miserable. It would have helped if you'd had a sling for DD, I think.
Incidentally, what do you mean about the vibe on trains? Were you travelling via national rail rather than the underground, overground and DLR? National rail is a different kettle of fish. The gaps between those trains and the platforms are insane in places - I'm always terrified as I jump down, or try to climb up. There is a very different feel to national rail than the rest of the network. Personally, I avoid it as much as possible.