You have to consider how long this property would work for you.
Are you planning on having more kids? If so, this house will become far too small very quickly. You will need a third bedroom, possibly in the loft - have the other houses on the street gone up into their lofts? Can you find any examples online of houses being sold on the street with a loft conversion and see what layout changes they have done to fit this in (i.e. where are the stairs and how much space does this cut out of the bedrooms below)?
The bathroom only being accessible through a bedroom is a problem and will continue to be so as your child gets older. As other posters have said, you can put a corridor in and reduce the size of the larger bedroom, which would help, but then you also have to think about a future loft conversion and how much space that will steal from that floor.
You could make it work while your child is small, but in about 3 or 4 years I would imagine you'd be looking to move again. There's not much you can do with this property to add more space, as you can't go sideways (I'm assuming the space next to the house is a shared drive), the garden is too small to go out the back, and going up only solves bedrooms and not living space.
My sister bought a house just like this when she first moved out of London, and within 4 years they had to move - despite a loft conversion - because though they had enough bedrooms after the loft was converted, the downstairs space was just too small for their family. They also had the one open plan room that the front door opened straight into, and my sister was driven mad with having nowhere to hide/store all the toys and no real dedicated adult space.
The question is, with the cost of moving being around 20-30k, and you doing this twice in the next 5 years, would it be worth pushing the budget a bit higher to get something that's more of a 10 year house now, and save yourself potentially 30k in future moving costs?
I don't know the area well but I know it's expensive. Think creatively - are there any ex council houses that represent better value for money? Could you go a bit further from the station and have a slightly longer walk in the morning? This is a very cutesy Victorian cottage that looks well situated from the map, and going for something less pretty and a bit more of a walk to the station could give you a much bigger house that could be a more longterm move.
Think it through carefully - if this is lower end of budget I'd be inclined to go a bit higher and get something a little more long term.