Bouncingblueberries, congratulations on your pregnancy ! I'm an allotment first timer so can't be much help but if it is just down the road that will help a lot I think.
What state is in your plot in and how much work does it need ? I think from what I've read it is best not to be over ambitious and just do a bit at a time. Potatoes might be a good idea to help break up the soil, apparently you get a smaller crop but it does make the ground easier to dig.
Wilkinson's have weed suppressing membrane which is £12 for a roll and should go a fair way over a half plot. What about lobbing that or some black plastic over, cut holes through it, dig out a bit of earth and plant courgettes and squashes ?
One of the guys at the allotment said he only ever waters new seedlings and his crops are fine and a friend of mine who had an allotment last year did much the same and I was surprised at how well things grew with little water so it sounds as it you can make hard work for yourself by over watering.
You can get purple podded beans and yellow mange tout which although would need picking really frequently, would be easier to see and might make a good job for a pre-schooler whilst you sit and supervise holding the baby.
Make a herb garden which should look good, be useful but not really need any maintenance. What about whacking in some raspberry canes, strawberry plants, gooseberries etc that you won't have to do anything to really.
It must seem daunting but if you can get friends to help with the getting it ready bit and take it step by step and go for easier crops and fruit bushes then I'm sure it will be fine. As the children get older and you have more time you can change what you grow.