Sorry OP, I got wrong end of stick about DC's behaviour... in your original post you said they were running in garden (and being told off by neighbour for running) so i got impression they were charging around. You asked if you were being unreasonable to 'expect her to put up with my DCs still running around'. I didn't realise noise-complaints were about noise inside flat rather than garden.
Differentname the OP mentioned several times she was worried DC would damage plants. She also stated 'I would rather it was still overgrown and they didn't have to worry about spoiling things'
This line stands out for me, because in a shared garden you DO have to worry about your kids spoiling things, and take steps to minimise damage. You can't expect a garden to be devoid of plants just to suit your young kids. Accidents happen yes, but if you're out there watching DC and they're quietly bug-hunting, collecting pebbles or playing hide-and-seek as you say, it's very unlikely anything will get damaged. Surely the only way they could destroy a plant is either stamping on it or ripping it to shreds? Which I'm sure they know not to do. If they stick to paths I don't see what the problem is. I don't think its reasonable to expect the garden to be free of plants just for your peace of mind. Could it be she's just planted seedlings and there are areas DC need to keep clear of until seedlings are more established? I can understand her being upset if a toddler is crushing seedlings underfoot, or if there are baby plants the DC are unaware of.
Having said that, I do think the way she's re-designed the garden is unfair and thoughtless, as it's taken away their playing area. I'm a bit confused as you originally said it's a large garden... has she dug-up the entire thing? Is there any grass left at all? If it's now one giant flowerbed, I suggest talking to landlord, explaining kids have nowhere to play, and asking if you can put down grass-seed or decking on part of it. It's possible he knew about her plan and approved it (in which case you just have to accept it) but it's possible he had no idea and might tell her to change one area back into lawn.
As I said before, try to claim an area for yourselves, get some garden furniture so you have somewhere to sit, toys like a slide or sandpit that stay in garden. Having your own things out there sends a clear message to neighbour that you won't be pushed out, that your DC will be using the space too. If there's literally no room to put your things, insist she moves some of hers or relocates some of the plants. However, I disagree with Alternative that you should remove plants without asking, bear in mind LL may have approved the new design. Legally she had no obligation to consult you if LL had given permission.
Alternative why should neighbour 'put it back to how it was', when previously the older tenant wasn't able to enjoy it in overgrown state?? Children are capable of enjoying a nice garden too, they just need to be more careful. It's still their garden to enjoy too, and the neighbour can't keep them out.
To all the posters telling OP to inflame the situation by giving kids noisy toys and water pistols... do you honestly think that will help??
The neighbour needs to be more tolerant and less precious over plants, but deliberately angering her and being rude will make things worse... it's likely to become a full-blown neighbour-war rather than a minor dispute over use of garden. Far better to have an adult discussion with neighbour and Landlord and try to reach a compromise.