A lot of how important it is to look round early will depend on how popular the private schools are in your area.
We started looking at private schools in April / May of DC1’s preschool year, after we didn’t get offered a Reception place in any state school we liked. We live in one of the poorer parts of the UK, so all the private schools within a reasonable drive of us still had spaces for that September’s Reception intake.
However if they are oversubscribed then the private schools near us give priority to who’s been first on the waiting list - all the ones near us are non-selective for nursery and Reception entry - so if having the option of going to a particular private school is important to you, then it’s best to register early to secure the place. Registering for a place on the waiting list doesn’t commit you to taking up a place at the school, so as a pp says, no harm in registering with several schools and deciding which you like the look of best later.
Our youngest DC was registered at their sibling’s private school within a month of birth, not strictly necessary as the private schools round here don’t fill up that quickly, but the registration fee was only £50 and it secures the school place.
Also, I’ve helped out with teas / coffees at school open mornings before, and it’s not at all unusual for families to come for a look around with babies or young toddlers. Lots of people like to get it sorted out early.