Sounds very exciting, to have a blank canvas to play with. Some photos would be nice but if it were me I'd be thinking thick hedging, with maybe butterfly bushes and/or verbena and achillea for height, which would bring birds/bees and butterflies to your garden, to help compensate for the noisy road. Also pots of all sizes so you could move them into shade. Will be lovely to see how you develop it!

Welcome to Goy! Plenty of scope there for perennials too. An easy attractive one is Lychnis coronanaria. Its in flower now, but when not in flower it has attractive silvery leaves. Easy to propagate if you want to. What's the hedge made of, because it can affect how many nutrients are lost from the soil to it (eg privet and leylandiis suck out a great deal ).
Penstemons seem happy in those conditions and even Nepeta (catmint) will have enough sun by 5.00pm. Rudbeckias ditto. A bright evergreen for year round interest is the low growing Euonymus fortunii, there are green/silver or green/gold ones and they don't mind plenty of shade. Spring bulbs are a must.
You'll get lots of other suggestions from other members.

Great ideas, Sunnydais and Steragram!
I'm not certain what the hedge is comprised of - mostly privet, but also something else which currently has small white of white flowers and which appears to be quite unhappy in the current very dry heat. I will try to upload some pictures. It would be difficult to remove it as it is against a low brick wall which doesn't belong to us (we are leasehold tenants of the local council) It would almost certainly need to be grubbed out with a mini-digger, for which there isn't the access. I think we're stuck with the hedge for now. I'm planning a walkway adjacent to the hedge so that we can maintain it, and this would enable us to plant away from the hedge so that they don't take all the available nutrients. Moveable pots are very high on the list, Sunnydais, and verbenas and achillea would really give height and dappled light, so not too shady for the neighbours. Perennials are welcome, Steragram, and I do like the sound of the green/silver varieties of Euonymus. I've never grown Penstemons, but have started looking closely at them this year. I like their resemblance to foxgloves and huge colour range. Thank you both so much!

Keep us posted with your progress!

I think you'll be surprised by the broad range of plants/shrubs & trees you can grow in your garden. For example my Autumn Twist Encore Azalea that I just bought yesterday needs part sun & part shade. It blooms 3 times a year and is the top selling azalea. That's one candidate.

But I would first look at the native plants of your area. The native plants would give you the least amount of trouble and wildlife depends on them. They should be available at the local garden center

Yes you could go for the cottage garden feel, the hedge will be a great backdrop, put in some perennials, larger to the back, medium in the middle and the lower ones to the front, but give each one enough room to bulk up and stand out, the secret is to plan your planting so that you have something happening throughout the year and not just in the summer, so you could opt for a mixed border and shove in some dwarf hebes, all evergreens in shades of green, gold, gray, purple etc, a small box ( Buxus) and some of the heaucharas and some black grass ophophogum to spill over the front along with a clump of Iberis, aubreitia etc, these will give interest through the winter, if you have room then cornus midwinter fire is a must, put in your spring bulbs in sept/Oct and voila your dream garden will take shape, give them a good start by incorporating some decent compost.

All these lovely ideas bouncing around for you! Good luck with your new garden. Make it a showstopper for your neighbours and passersby!
The white flower you mention could be from the privet hedge itself as it will flower if allowed to. Doesn't smell nice, though!

A matter of taste - I love the smell of privet...each to his own!

Funny that, Stera! I would have agreed with you a few years ago, but now I dislike it.

I know a lot of people do dislike it. I hope don't change into one of them...