Not very, and the numbers do not pretend otherwise. Pucklechurch has no railway station of its own, and effectively nobody here commutes by train. Buses exist but barely register, carrying only around one in fifty working residents to their jobs. Just over half drive everywhere that matters, and a third have sidestepped the problem entirely by working from home. This is a village you leave by car, or you do not really leave.
On public transport the honest word is thin. There is no station in the village, and the share of locals who get to work by rail rounds to zero, which tells you how realistic the train is as a daily habit from here. Buses are present in the geographic sense, with around 20 stops dotted about the place, but only about one in fifty residents actually commutes on them, and a stop is not a service when the timetable is sparse. The motorway runs close by, yet there is no junction here, so the fast road everyone else relies on is scenery rather than access until you have driven a fair way to join it somewhere else.
Broadband is the one line where the news improves, provided you mean the digital kind of connection. Superfast speeds reach every postcode in the village, so ordinary streaming and video calls hold up without a fight. The top tier lags behind, though. Full gigabit connections cover only about seven percent of the place, which makes the fastest available broadband a minority sport here rather than the default. For most households the everyday connection is more than adequate; for anyone chasing the very quickest, the coverage has not turned up yet.
What all of this means in practice is straightforward. If you own a car, Pucklechurch works, and the roads quietly do the job the buses and trains decline to. If you do not, the village asks a great deal of you, because the alternatives to driving are so slight they barely add up to a network. The households that thrive here are the ones who already intended to drive, or who work from a spare room and treat the commute as a stroll to the kettle. Everyone else ends up spending more time organising travel than the map would ever suggest.
So the verdict is blunt. On connectivity Pucklechurch scores poorly, propped up a little by solid everyday broadband and let down by public transport that is closer to decorative than functional. Buy here on the understanding that you drive, because the village does not hand you a serious way to avoid it.