You haven't used the "oldlogon" field for intranet.dbo.members in your query.
database 1, table 1: id, newlogon
database 1, table 2: oldlogon -- I need to update this field
database 2, table 1: id, oldlogon
database1, table 1: intranetnsops.dbo.tempmembersonemail
id - employeenumber
newlogon - newlogin
database1, table 2: intranetnsops.dbo.rb_booking
oldlogon - owner
database2, table 1: intranet.dbo.members
id - employeenumber
oldlogon - ???
I assumed that the field name for "oldlogon" in intranet.dbo.members is "owner". If not, change the field name accordingly.
UPDATE T1SET owner=T3.newlogonFROM intranetnsops.dbo.rb_booking T1 INNERJOIN intranet.dbo.members T2 ON T1.owner = T2.owner INNERJOIN intranetnsops.dbo.tempmembersonemail T3 ON T2.employeenumber=T3.employeenumber