Yes the district council approve the street name changes (or new ones) and forward this to OS and Royal Mail who usually update records within 3 months, it can take a long time to reach Google and sat-navs though!
Interesting, that's a newer development so I could dig out the records and see the reason given for naming it Tithe Barn Gardens. There's The Tythe in Midway too, which is the only other example of that spelling I could find.

Good point on the houses; we've found there are two street name plates with two spellings (Tithebarn and Tythebarn) at each end of the road (misspelt name plates is a surprisingly common issue in South Derbyshire and is being corrected where feasible). The street naming officer has advised that the road is named "Tithebarn Lane" based on their research.