Unless they have found an issue which needs a part they may not have in stock expecting you to wait an extra day is unacceptable - a service only takes at most a couple of hours.
You can go outside of dealership, but check what savings you actually make and you will also need evidence that all the parts fitted are the same as would have been fitted at the dealership.
You also lose any potential for a goodwill repair just outside of warranty.
I benefited from this last year - a part failed which wasn't covered by warranty, but as I had full dealer history it was done free of charge - it saved me £350.
Result, but not without a telling off from OH, who would not have taken it any further, next year they will pick up and drop off the car and we have a free MOT. Perhaps they will do something wrong to the car after the call from Head Office.