Can anybody suggest a quick growing shade loving flowering climber to go around a trellis archway

How much shade?
By their very nature, climbers tend to like sun.

How big is the container? If you want a fairly tall climber, it's going to have quite a large rootball. Honeysuckle likes shade and most varieties are scented

The spot only gets about 3hrs direct sun per day in Summer. It only needs to cover about 6ft in height.Container can be as big as required,within reason.

So the containers will be moveable then?
There is a winter flowering Clematis (which Andrew may be able to help on as they're not really my thing).
For summer, may I suggest a couple of exotics. Gloriosa Superba (cut and paste the link below) is very easy and fast growing. They tolerate shade and can be bought into the cupboard under the stairs for winter. The tuber multiply reliably too.

http://www.growsonyou.com/garden/show_garden_item/25770-gloriosa-superba

My second suggestion for summer would be Lapageria rosea. Not cheap, but sure to make an impact! Again, can be bought in (conservatory) for the winter.

I had a lovely dark purple clematis called 'Black Prince' in a spot like that last year, and it did brilliantly and grew fast.

Meanie - were you thinking of clematis cirrhosa? That goes dormant in the summer but is not hardy in very cold areas. If you are swopping containers, you will have to unravel your climbers from the trellis every time.
If you can find a supplier, there is a dwarf form of the golden hop called 'Golden Tassles' with attractive leaves and hops in the autumn - goes dormant in winter and starts again each spring (don't get the ordinary one, it will be much too vigourous)