Rosie - not a running club as such, but my wife and a few other local mums try and go running every Wednesday evening. They tend to meet in Shad Thames and do a few 'laps' crossing various bridges. If you're interested then PM me and I will give you my wife's mobile number. It's an open thing - anyone welcome to join them I think.
Not sure if you're willing to venture north of the river, but the Sweaty Betty running shop does a running night on Tuesdays - meets at 6:15, we do runs from the shop around the area, usually covering around 5-7 km plus drills, stretching, etc. I'd highly recommend it, the group is friendly, ranges from 10-18 ladies of all levels. The shop also does a nice promotion for new participants - if you come to the class 5 times you get a complimentary shirt of your choice (up to a certain amount). After 10 runs you can buy a half price pair of trainers!