You never said what type of plants you wanted but here are but a few -- hostas - Ferns - columbine - Ivy - Periwinkle -- Ken

Now's not a great time, its a bit late, September/October would have been best for most plants (though not all). Weather forecast indicates arctic conditions coming in overnight even here down south...
How big is the eucalyptus, or how long has it been in situ?

Thank you Ken bonsor for your ideas. The area is quite small, and in a raised bed, which already has periwinkle and some ivy, which I try to keep down. What I was thinking of was something that would literally spread rather densely along the ground without trying to climb. Hostas get eaten to bits by slugs here (wet clay soil). I haven't tried ferns.

Thank you, Bamboo. The tree has been in situ for 20 years, and is huge- taller than the house - with a lovely shape. It sheds its bark in slivers, and drops little pointy fruits(?) in the summer, as well as flowers which have a beautiful pungent smell. So....a bit tricky, but we are fond of it and work around it.

This one doesn't spread out all that much but it would look lovely under your tree - there is a pulmonaria that has very big leaves spotted with white, and has blue flowers in spring. Sorry I don't know the variety as mine came from a plant sale. Looks great interplanted with smallish daffodils. Not if the soil is very dry though.Then you could have some ground cover perennial geraniums in between - some are evergreen and quite low growing. sa
All these would look nice with your ferns too.

Epimediums interplanted with spring and autumn flowering cyclamen

Funny, Badfish, I was looking at one today and thinking it would have been worth a mention with its lovely shiny leaves.

Having read your further info in your response to both me and Ken, I'd suggest Cotoneaster 'Queen of Carpets'. These tolerate dry quite well, but you will need to plant away from the base of the tree so there's enough soil for the plant to put out roots. They prefer a bit of sun, so you might find the plant will spread towards where the light is brighter rather than going back towards the tree. This shrub has the habit of spreading out across the ground, rooting as it goes. Bit slow to get going initially, keep watered the first couple of years.

Many thanks to Steragram, Badfish and Bamboo for some more lovely ideas. I have made a start and got the ferns in and the spring bulbs. Need to get to a good garden centre for the others, but might still make it as the sun is shining and the temperature's still OK down here in Greater London.

Weather's OK?! It's blooming freezing... thick white overnight, had to scrape the car windscreen, hail, rain, snow, we've had the lot today in West London...;-)

Bamboo, yes, perhaps I sounded a bit optimistic, but I did some more planting while it was sunny, though it was cold. No frost either on the car in the morning, or on the ground, though we had a big downpour later in the day. The weather seems quite localised.

Yea, you're right - someone only 4 miles away didn't have any hail or snow when we did.