I do believe the agapanthus loves full sun and I would move it when the leaves start to die down...around early november when there are no frosts.
I have always grown my agapanthus in pots. I find them hardy and for the first time in years I moved the pot into a cold greenhouse last winter...not really sure why as they have always come through the winter unscathed.
I know they thrive and flower best if their roots are pot bound. Mine is a variety called Northern Star a beautiful dark blue. It is in a 15 inch diameter pot and this year had 24 blooms...no kidding. I also know they love to be watered daily in the growing and flowering season.

We had late frosts this spring and that affected those Agapanthus outside - certainly mine were burned by frost and that pushed flowering back this year. Full sun is certainly the way to go.

Sorry Johnp, gonna give slightly different advice, no offence!
You can move them in the next 3 weeks, while the soil still has some warmth, don't wait till midwinter, but water them well after moving and keep watered if the weather is warm and dry. Ours were late flowering this year because of the cold spring and early summer, I've only just removed the spent flower heads, and they're not in full sun either. As for keeping the roots cramped, that is the received wisdom, but frankly, I find the hardier ones flower much more prolifically in open ground, unconstricted, than they ever do rootbound in pots.

In South Africa they are often planted in mounds for landcaping so people can see the flowers better. Might be a good way to try in the UK with the higher rainfall, esp in winter.

Many thanks for your answers all most helpdul