I've been using Moss Tyres on Clifton Moor recently - handy for my work and seem to be always busy. Had them recommended by the MOT Shop on James St. Not sure they're the cheapest but seem OK:
Not a first choice (IMO) is Swift just off Foss Islands Road - went there initially as they advertised as an independent Citroen specialist, did the job but definitely pricey:
A good cheap back street option is Viking Vehicle Services on Bull Lane off Lawrence St 01904 426756 - don't be put off by appearances. They (Mick) did a good and very budget job with my girlfriend's Punto for several years.
No experience of any others.