Mw - I purchased two small holly bushes here, from a flower shop! One a variegated, cream and dark green and must be a female as its flowered and has berries and a plain green one, which luckily is the male. With Holly you do need, either in your garden or next doors a male tree, or no babies I mean berries.
Holly is slow growing, till they take off then need a bit of pruning , which you can do at Christmas when you bring in the holly Its the evergreen bit that is causing me problems. You say you do not like conifers. But there are some beautiful species, shapes and forms, some carrying from green to violet green cones. You may have to be content with cover from climbers and small trees and shrubs from early spring till Christmas, then hopefully you will get the usual snow and will not be able to see far anyway!!.
My choice would be a climbing Hydrangea, honeysuckle,a clematis or two, Partenocissus tricuspidata, or Japanese Ivy, beautiful red and crimson leaves in the Autumn. This will cover large expanses. In front of these perhaps an evergreen Elaegnus, yellow and green leaves. Cotoneaster and Forsythia. A Berberi. , they are evergreen too. Aucuba looks like a spotted Laurel, evergreen. Cytisus or broom.
Then towards the front smaller shrubs and perennials.
There will be nothing like an instant hedge or garden come to that, but the sooner you get them in the ground the sooner you will get results. As for your privet, put them in a spare piece of dug over ground, weed free about eight inches apart, keep watered, provide a bit of shade may be. When they get to a size you can actually say you have a bush, not too big mind, about 15-18 inches, dig up with plenty of soil and put in final position. Do this in October.
Trawl the internet, get an idea of what trees and shrubs ect you like then keep an eye out to see if they grow local, or ask on here. Get to Nurseries and plant centre's at different times of the year to see what is in leaf or flower. Gardeners are usually a generous lot, especially when you take an interest in their gardens, they like to share so get making some gardening friends, thats my advice.