Saw the queen 10 days ago...

[Angularity]

Are you certain your colonies are queenless? Have a good look at them (carefully) and watch out for emerged queen cells. If there are queens in your two colonies, you could add your newly purchased queen only to have the colony kill her as they are already queenright.
The swarms in the neighbouring gardens do not necessarily have to have come from your two colonies. Are there other beekeepers in the area? Feral colonies? I would suggest that you try test frames in your colonies to see if they try to draw out emergency queen cells. The help of an experienced local beekeeper will be invaluable and will help you learn a wee bit more about practical beekeeping.

They're definitely queenless, no capped brood at all, no eggs, no larvae, bees busily back-filling what was brood with honey.

I think the first two came from my colonies, based on how many bees I have now, but the second two came from elsewhere, based on bee colour, number and behaviour (mine are/were placid, the third and fourth were aggressive, as in an experienced beekeeper running down the road and finding a tree to hide against aggressive).

 

[Teemore]

Queenless or just no laying queen? Test frames will help determine whether or not they are queenless.

 

[madasafish]

They're definitely queenless, no capped brood at all, no eggs, no larvae, bees busily back-filling what was brood with honey.

I think the first two came from my colonies, based on how many bees I have now, but the second two came from elsewhere, based on bee colour, number and behaviour (mine are/were placid, the third and fourth were aggressive, as in an experienced beekeeper running down the road and finding a tree to hide against aggressive).

Most of my hives are very placid. Even when Q- they are placid.

Test frame should help - a lot..

New queens are hard to find and can take 3 weeks to start laying.(polished cells are a clue)

 

[jenkinsbrynmair]

Queenless or just no laying queen? Test frames will help determine whether or not they are queenless.

Not having any brood indicates nothing. Could be a Virgin in there somewhere

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[REDWOOD]

Fanning on the top bars is a good indication there is no queen especially if you haven't seen that before with that colony, but don't confuse this with queen away on mating flights

 

[Erichalfbee]

Fanning on the top bars is a good indication there is no queen especially if you haven't seen that before with that colony, but don't confuse this with queen away on mating flights

Yes.....I've seen that