Ok, well, as well as local activity, the tram gets you into town quite easily, so you do have options.
With a one-year-old I expect you'll have found the really obvious things like the local library (mine loved Rhyme, Rattle and Roll at that age), parks and children's centre - worth looking at. A couple of links if not:
local.mumsnet.com/nottingham/libraries/158424-hucknall-library
local.mumsnet.com/nottingham/childrens-centres/162124-butlers-hill-and-broomhill-childrens-centre
We need to add some playgroup listings for your area, which is on the to do list but if you come across any good ones first, do add them or flag them up to us.
In town, a couple of thoughts are that with an almost one-year-old, you can still go to the 'Bringing Up Baby' screenings at Broadway Cinema: local.mumsnet.com/nottingham/cinemas/152775-broadway-cinema
The free family workshops at Nottingham Contemporary are good for any age, the galleries are worth a look (also free) - and the cafe is really good for small children, spacious, relaxed, with plenty of highchairs.
They seem to have paused their regular weekend workshops during July, until the school holidays, when they're running every day except Mondays. So, do keep an eye on our Summer Holiday listings. While lots of stuff is aimed at school-age children, there is a lot on for families too.
Otherwise, Nottingham centre is quite compact and easy to explore.
To learn more about the area, there are quite a few museums locally, listed in here: local.mumsnet.com/nottingham/attractions
Also in that section, Stonebridge City Farm might be fun, certainly in a few month's time. It's not far from the city centre, maybe a bus ride though, depending how much you like walking.
Hope that's some help as a few quick thoughts! 