OP, now that you've given us some more info, some further thoughts which may be helpful.
There is a charity called Pets as Therapy - they have dogs which are carefully assessed as being super docile, gentle and good with kids/older people - the dogs are often taken in to visits to schools and old peoples homes. You could contact them and they would be very happy to provide a supervised meeting for you and your DC to allow your DC to meet a gentle, well behaved dog and slowly become accustomed to them. As your DC grow up they will not be able to avoid meeting dogs in the street, or at the park, so it makes sense to try to work to avoid a deep fear.
There are also some brilliant American posters, produced by a vet called Sophia Yin, which give lovely, clear instructions on how children and dogs can interact and play together. You can download and print them for free - see link [[http://info.drsophiayin.com/download-free-poster-how-kids-and-pets-should-not-interact/ here]]
For the puppy, a few more ideas. It's good that you've contacted the Dog Warden - you haven't said if the roaming has continued? If it does so, please get back on to the DW as soon as possible. It will also be worth contacting your local Councillor and insisting the Council take action. If you are in social housing, please also report the dog straying to the local housing office - it could well be in breach of tenancy agreement. Many social landlords are now making chipping of dogs compulsory ( a very positive step) and recognise the problems caused by irresponsible dog ownership. Ensure your own fences/garden are secure so that it cannot stray into your garden area. If you have a PCSO in your area, contact them as well and make sure you report the straying via local non emergency number too, as it's recognised now as being Anti Social Behaviour. The key with many of these public agencies is to insist on complaints being taken, as often local priorities are driven by numbers of complaints, so make this work for you. It may well also be useful to attend a local PACT meeting if problem continues.
While I do sympathise with your concerns, I'd say that the puppy is at far greater risk at the moment - by straying, he/she could be hit by a car, stolen, poisoned, or chased by other animals. Also, if picked up by DW, the charges for release are quite high.