It's a pity Nutcracker seems to have taken his leave of BF, he'd have given you a much better answer.
There are a couple of city centre parks - Leazes Park (near the footy ground) and Exhibition Park near the University / Hancock Museum, but I've not personally seen nuthatch there - they're both relatively open parks and relatively isolated within the urban area. You'd probably have more luck visiting Jesmond Dene / Heaton Park area in the Ouseburn Valley just east of the city centre, which have much more extensive mature woodland. For an absolute nailed-on certainty though you need to go to Gosforth Park Nature Reserve - it's owned by the Natural History Society, you pay an entry fee to the warden as a day visitor - the entrance is by a lodge which is located at the SE corner of the site.
For getting around if you're using public transport, Newcastle is now covered by the City Mapper app, which will give you plenty of options.