Yes, but Locks opening and closing times vary... by season, ie: April-October and from November-March.
No swing bridges are operated on the weekends or holidays.
Marsh Lock is always open for pleasure craft (but you have to have prior clearance and paperwork for the Manchester Ship Canal.
I know that The Mersey is not privately owned... In fact, don't even know where you got that idea from... it is formed from the three tributaries of the Rivers: Etherow, Goyt and the River Tame; becoming the Mersey near Stockport, Cheshire.
I remember however, that in some spots on the River Weaver, the width narrows to about ten feet, and in some areas, (the moorings at the Riverside Inn, for example) are too shallow for narrow boats.
So. It all depends on the type, width, draft of the vessel you are in, as to how wonderful (or miserable) your voyage may be... In some areas, boaters are being discouraged to leave the vessels, because of vandals... But... what a trip!
You will (a must have) need a current navigational map. However, the site below has updates as to reported conditions...
From Stockport it flows near Didsbury, Stretford, Urmston, Flixton, and then at Irlam it flows into the Manchester Ship Canal, which canalised the River Irwell to this point. The course of the Mersey has been changed by the Canal past Hollins Green to Rixton. At Rixton the River Bollin enters the Canal from the south and the Mersey leaves the Canal to the north, meandering through Woolston, where the Ship Canal Company's dredgings have formed a nature reserve (Woolston Eyes), and Warrington. It is tidal from Howley Weir in Warrington, although high spring tides often top the weir. At the Runcorn Gap between Widnes & Runcorn rail and road bridges (Runcorn Bridge) span the river and the Ship Canal, which runs alongside the widening estuary to Eastham Locks where canal and river unite, and from here the estuary narrows to flow between Liverpool and Birkenhead into Liverpool Bay on the Irish Sea, after a total course of around 70 miles. The conurbation on its banks near the coast is known as Merseyside. The eastern part of this estuary is much affected by silting, and even marked on updated maps as dry land instead of as tidal. At the point between the Woodside Ferry Terminal and Albert Dock, the Mersey is about a mile wide.
Have a wonderful trip!