Having got rid of large conifers, what shrubs will happily grow instead?

Crumbs....thats a good question....it depends on the size of the border and what sort of effect you want. However, the first thing you will need to do is feed up the soil. The conifers will have taken all the moisture and goodness out. Dig in plenty of well rotted manure, soil conditioner,,,whatever you can get hold of ...then plan your border.
Good luck and welcome to GoY.!!

Hi Horsey, welcome to Goy.
You will have to replenish the ground as Amblealice has said, where the conifers were growing. You could plant Choisya ternata, Cornus, Escallonia, Ceanothus or Forsythia, to name a few.

Hi Horsey and welcome... as Dr B says the soil will need a lot of feeding. Compost, composted bark, peat (if you use if - we don't), spent mushroom compost anything that will add to the soils vitality. I am assuming that the roots of the conifers were removed when so it really does depend on the size of the border and whether you have acidic or alkaline soil. If you have an acidic soil Acer palmatum will look wonderful with the sun through it so long as you can keep its feet cool, as will Pieris japonica. You could also go the bamboo, grasses route... in fact the possibilities are just about endless! Wow I almost envy you :-)

Try and plan a succession of flowering times and put in spring bulbs around the base of them (available soon). Try Abelia grandiflora (long period of colour), Potentilla 'Abbotswood' or the yellow one, Phlomis, some red stemmed Cornus or willows for winter interest, Philadelphus...how much space have you got?