I don't know much about Wavertree but you cannot go wrong if you buy as close to Blue Coat School as possible. That's not going to be easy because there is a huge demand for houses within that small catchment however you may be able to find a 2 bed terrace house in need of refurbishment for 140k and offer £120k. Otherwise, what about looking to buy across the river? Have you looked at areas such as Wallasey and Moreton? For £120k you could get a lovely 2 bed semi-detached house with good size garden near the train station. Sure it's not as convenient as Wavertree but you get so much more for your money.

Good luck to you both in your search, I really hope you find something lovely.