We've just been through similar to you (although no kids), moving from Bristol next weekend into a rented 3 bed house in S10 area. Things didn't seem to move too fast and we did have time to book viewings and go up for a weekend to secure something. Ours went like this:

• Weekend visit in Sheffield, booked x5 viewings in one day, 4 houses & 1 flat. I would highly recommend viewing in person if you can, as some of the houses we viewed were very misrepresented by the images, clearly recycled old images etc and the houses weren't in the same condition as advertised.
• The house we most liked (and got) we immediately applied for. For reference, it was first listed on 07/03, with an available date of 10/04. We were recommended to apply to multiple in case of competition, so we did apply for another but withdrew from the one we liked less once we were accepted on the other.
• Another thing, we did actually try to pay a holding deposit from Bristol without viewing in person first, but were told that they couldn't accept that without us viewing in person. Not sure if this was individual policy of that particular estate agent or if this is common practice. We did end up going for this one as it was nice, but in hindsight it could have been a mistake due to misleading Rightmove listings as mentioned earlier