Hi both, sorry for late reply, I thought I'd no takers. :)
Iamyourequal - you sound all sorted. It's great that you are taking lessons. And yes, I totally get that roads are a bit daunting. Firstly remember that folk on bikes have every right to be on the road as much as any cars. And though it's totally understandable, cycling on the pavement is illegal (though with a kid I don't imagine you'd ever get done for it).
Familiarise yourselves with the highway code for bikes but better still, get a book on the rules and tips. I'm in Scotland and we have Bikeability here (which is basically Cycling Proficiency with a new name). You'll probably have useful local stuff from your council depending where you are but otherwise here www.cycling.scot/cyclopedia/nursery-school?filter=resources is a good place to start as it has lots of resources.
Start out on quiet roads first. Two key words to remember are CONSPICUOUS and PERSPICUOUS. The first is obviously taken care of with hi-vis clothing, however the second means make it obvious what your intentions are. So be confident in holding your line, with your signals, especially when turning right. Also remember it is perfectly legal to 'take the lane' which is sometimes referred to as riding in the primary position. There is no obligation to cycle well over to the left, most folk on bikes do this as a matter of courtesy.
In urban environments, it's safer to ride in the middle of the lane which makes you more visible.
In all honesty, I used to teach kids that were aged about ten but I felt that it was in one ear and out the other, especially if they never went on their bikes from one lesson to the next. The only ones who really improved were the ones with parents like yourselves, who put the time in and went out cycling with them.
Allow your child to cycle slightly ahead and you cycle behind them and out a bit, to give them a bit of protection and comfort from other road users.
The vast majority of vehicle drivers will be obliging when they see kids out on the road but without over stating it, there are always one or two twits who pass a bit on the close side. Hence the reason to ride out from the pavement a bit, it means that they won't pass unless it's completely clear on the other lane (which is what they are supposed to do anyway). Also means you give yourself more room to avoid dodgy drains etc.
Sadly the modern age means car is king and the UK has a long way to go to catch up with the likes of Holland & Denmark but go out a little each day. This will gradually build up confidence.