Interesting.
My father-in-law lived in your quaint old house after having converted it from the coach house that served the large house on the corner of Honor Oak Road. I reckon the proposed side extension to No.1 is very in-keeping with its surroundings but, in Planning terms, my guess is that it will fail at least on the grounds that it rather impinges on your place. It appears that the gap between the 2-storey flank walls will be about a metre apart, which would tend not to be allowed in a normal street scene let alone in this particular Conservation Area. Having lived in the place for a few years myself, I think it would be a bit of a shame if you become crowded-out by the proposed development. When faced with these types of situations it's always best to prepare one's self for the worst, but at least here you're backing the fight with your petition, which you'll no doubt use as a tool to get the backing of local conservation groups and Councillors who may in turn be able to assert some pressure on the Planners' before their final decision is made.