Yes, they did. Priority was always to the busiest places, i.e. main roads, main shopping streets, outside schools/hospitals, etc. But as the days passed on the long cold spells (like we're suffering now), they'd work their way through other areas too.
I used to live on a quiet residential street, no shops, no school, no GP surgery, etc. We'd usually get a gritter wagon sometime during the day on day 2 of an icy-spell, and we'd get the pavements gritted usually around day 3 or 4.
The town centre roads would be gritted several times overnight and the pedestrianised areas/pavements with shops etc would have been gritted first thing in the morning before most poeple were out and about.
They used to re-allocate workers from other tasks, such as bin lorry drivers would drive the gritter trucks, bin men and gardeners and maintenance workers would be on the vans shovelling grit on pavements, etc. I don't think they do that anymore - certainly, normal bin collections have continued around here over the past week, with the bin men slipping and sliding all over the place as the roads/pavements havn't been gritted. Presumably it's the unions and their "one man, one job" approach which means council workers can't be redeployed.