Morgan,
Yes, Wallsend town and wider parish was covered by Tynemouth Registration District from 1837 ( A wee bit quirky in that Register Office was in town of North Shields- even Tynemouth Parish Church lay in town of North Shields !)
Tynemouth RD covered a wide area in south east Northumberland, including Wallsend ( lying between Newcastle and Tynemouth/North Shields)
Tynemouth RD changed it's name in 1936 to Northumberland South.
With the creation of the new county of Tyne and Wear in 1974 out of the southern bit of Northumberland ( and a north bit of County Durham)
the same Reg Office divied up it's records, some to Northumberland Central and Newcastle, and get the rest and got a new name of North Tyneside Reg District.
Sally, I was surprised to see that on 1901 census, there were only 10 Reidy folk in all of Northumberland - all one family under Ireland-born Michael, a Customs official, and Ellen Reidy. Living in the town of North Shields. (Census page ref 4603-77-10))
Also on 1901 census ( 4796-93-30)) was Elizabeth Lawler, 13 yr old dtr of Ireland-born widower Thomas, living in Vine St, Wallsend.
Michael
Newcastle