Scotland has unrestricted Sunday trading, and has had it for many years. What happens in Scotland with the bus services there?
Generally, we still have a much reduced service on Sundays, with quite a lot of routes still not operating. For example, the buses that run outside my door run at least 5 times an hour Mon-Sat, reduced to just two on a Sunday. The start times are much later too.
We've never had restricted Sunday trading in Scotland (at least, not in my lifetime), so I can't comment if there's any increase in demand for public transport as a result. I suspect that demand will increase slowly over time.
One thing I can say is that Sundays are busy shopping days in Scotland, and it's fantastic to see. Trains into Glasgow on Sundays are usually close to capacity. Buchanan Street can easily be as busy on a Sunday afternoon as on a Saturday. Shops can turn over a lot of money on a Sunday. When I worked for a supermarket, I worked most Sundays for a few years, and we were certainly busy for more than six hours a day. In fact, given how busy our store was, I'm glad we were open almost as long on Sundays as we were on Saturdays - if we concentrated all the trade onto Saturday we would have been overwhelmed.
Sunday has the potential to be a big shopping day if the laws in England are relaxed. It will not be a sudden change, but once you start to get a lot of shops opening for longer, you'll start to see quite a marked increase in footfall I suspect. And with that, I would imagine bus services may see an increase in demand and, subsequently, some increase in frequency on some routes.