Clare, there is no doubt that running more miles per week (sensibly) would help you improve the time but with limiting running time available you would probably gain more by doing one of your runs faster. If you can't join a club, there are, as has been said, plenty of published training plans. Most of them assume that you have plenty of training times available so you will need to customise to suit. Generally, running less than the plan suggests will not cause too much of an issue, but be wary of trying to make up the deficit by running too hard or too fast - it may work for some, but most of us just end up knackered and fed up. In my opinion, some sort of threshold type work will be the most valuable for you now you have the basic endurance to go the distance.
The canal system is definitely good for daylight runs. Sutton Park is a good running venue but the Clent Hills will be a bit nearer for you - also good. You could try heading down the A38 Bristol Road to the Lickey Hills - not a bad run in itself but well worth the drive for a relaxing countryside run. Canon Hill park isn't bad and it connects to a riverside walk paralleling the Pershore road to Stirchley, you can then turn left on the number 11 route (Fordhouse Lane/Vicarage Road) up to Kings Heath Park and Highfield park, leading back to Canon Hill Park along tree-lined streets.
I ran the first one and found it to be a great event. If you are not used to hills it will be hard, but it is very scenic - you leave Basingstoke quickly and don't come back until near the end. Yes there was a medal.
Km. I use metric for all measurements. Miles/km is the one conversion that I will do readily, especially when talking to folk who still work in the old units. Converting pace and distance is a useful mental exercise when languishing through the last 5k of a half.