will tubs of bizzy lizzies grow every year when they are kept out of the frost in a greenhouse

Unlikely, Shane. They would get 'leggy' as they need the brighter light of spring and summer as well as warmth. Better to start again with a fresh lot, really!

Spritz has summed it up for you.
Some will survive and some will not.
They are normally grown as annuals.

if you particually like the variety you have why not try collecting seed, can't say i have ever done it myself, but would imagine it is possible to collect and then re sow, but thinking about it, would imagine no guarentee that the vareity will be the same anyway due to cross pollination, anyone tried this with Bizzy Lizzie?

I have with the singles, and not had a problem. This year I have saved some doubles that I kept well away from the singles. so cant wait to see the results

I've collected seeds too from a salmon pink, white and lilac colours and will start them off in the greenhouse early next year. I have had experience (in Essex) of self-seeded plants where they were grown the year before, but they were always very late to appear and never had time in the season to grow bushy, so they obviously need a helping hand.

You can root new cuttings from a busy Lizzie,break off a few stems and place them in a jar of water, it takes about a week and each stem will have grown new roots and can be potted up. When grown a bit pinch out the top to encourage it to bush out. this works with tomato sideshoots too, same method. I only ever buy one tomato plant and make new ones from the sideshoots