Hi Aimi - congratulations on your move! We've been in Tunbridge Wells for just over a year now with DS who is 19 months and it has been so lovely. Best move we've made I think!
I am not in St James area but just wanted to let you know about a couple of the things we get up to:
Wednesday Mornings (9.45-11ish) there is a soft play session at the sports centre on St John's Road. This is absolutely brilliant - bouncy castle, climbing frames and slides etc ball pit, all aimed at little ones and usually not too many loopy big ones tearing about so a great place to let your 14mo burn off some energy! It is in one of their big sports halls so hey have so much space to range about. There are lots of parents there with one toddler running about and a newborn snoozing in the pram/carseat at the edge so you would be in fine company when your next little one arrives :)
Most of the churches do a playgroup - over your end of town I have heard good things about the "bikes and slides" group at St Barnabus church. Most of these are fairly low-key religious so if you aren't a regular churchgoer you shouldn't feel uncomfortable!
Other than that St James' Park is brilliant and usually packed with mummies and toddlers so I am sure you will find some friendly faces there. Dunorlan Park is also lovely and the Grove in town is great too.
I feel like a bit of a MN traitor saying this but the Tunbridge Wells Netmums group does a meet up every wednesday at different locations, this is how I met most of my friends here, they are so good (you get a name badge and everything
and they are so welcoming! Have a look at their facebook page here
NCT runs a "bumps and babies" group at the Camden Centre in town on a Tuesday afternoon, but I think that is for littler ones than yours - it is an excellent place for information and a cup of tea and a chat though and they do loads of taster sessions from the local music groups etc.
I wasn't in TW when I had DS but I hear that there is a brilliant ante-natal yoga group run at the Camden Centre. Also lucky you - the new hospital at Pembury looks like an amazing place to have a baby (if you are opting for hospital that is...)
Sorry this turned into a bit of a rambling message, hope you enjoy your new home 