It's so interesting what people's priorities are. In London you have to distinguish between absolutely essential, would make your life easier and would be nice to have. And to realise that these change as your children grow. We have been in our house since our children were tiny and now have teenagers.
I would put schools at number 1 as an absolute essential. If you are planning to stay put for a few years then you need to look at the secondary options or be prepared to move again when your eldest is 9 or so.
After that a garden of some sort would be next on my list. With a baby and a toddler it can be remarkably hard to get out of the house sometimes, baby is asleep, toddler won't get dressed, you're expecting a delivery etc. Children need somewhere to be noisy and run around without driving you mad and so do their visiting friends.
I wouldn't put parking as a high priority myself. I've never had off-street parking and it doesn't bother me. See if there is residents only parking to keep the spaces free from commuters etc, in the context of house prices the cost of a permit is a drop in the ocean and and if you can park outside the house and get the children out on the pavement side it's fine.
We have one bathroom and it's fine. Downstairs or second loo is nice to have especially if potty training, but potty training time is finite. I always had a potty close by when the children were learning anyway so loo close by is not essential I think.
I would want a bedroom each for the children. They will need their own space when they are older.
Near a park is nice (and what a lot of friends envy about our location). Great for letting off steam, picnics and learning to ride bikes.
Near a railway station or tube station and other public transport links is good to ensure you and your DP don't spend your lives commuting and great for day trips from quite an early age. I love taking my children into London and being able to get there and back quickly and easily.
Spare room for visiting grandparents and friends is nice to have.
I agree on the hall space for the double buggy now and trikes and bikes later.