Couple looking to rent 2 bedroom property

My husband (Alastair) and I (Catherine) are searching for a 2 - 3 bedroom property to rent with our cat, Luna. We would be renting the entire property and not a single room. Alastair and I are both working professionals and would like to find a place within reasonable walking distance of a bus stop or train station for that reason. Preferably as close to the Jubilee line as possible as we both catch it to get to work. However, we do not want anything directly next to a busy road or train track simply for our pet’s safety. We would prefer a place that has a bathroom with a shower and a bath (a bath/shower combo is fine); kitchen with a washing machine and dryer; and a small closed-off garden (for Luna). We need one of the rooms to accommodate a queen size bed and the other room should be large enough for a desk and a single bed. Both of us are very neat and tidy people; we don't smoke and we don't take recreational drugs. Luna is herself very clean (cats are incredibly fastidious creatures) and you do not need to worry about her messing up furniture or carpets, etc. We are more than happy to send any interested landlord/lady a pet resume and previous landlord references. Any indication of black mould and water damage is unfortunately a deal breaker for us as I suffer from asthma.