Hi RichardSa,
Refer below query.
CREATE TABLE Users2
(
	Id INT
	,email VARCHAR(50)
	,Name VARCHAR(50)
	,CreatedBY VARCHAR(60)
	,LastLogin VARCHAR(70)
)
INSERT INTO Users2 VALUES('1', 'richard@gmail.com', 'RichardSa', 'richard@gmail.com', '14-Feb-23 8:05AM')
INSERT INTO Users2 VALUES('2', 'sammy@gmail.com', 'RichardSa', 'richard@gmail.com', '17-Feb-23 1:35PM')
INSERT INTO Users2 VALUES('3', 'manny@gmail.com', 'RichardSa', 'richard@gmail.com', '20-Feb-23 6:48PM')
INSERT INTO Users2 VALUES('4', 'mike12@gmail.com', 'Mikey', 'mike12@gmail.com', '20-Feb-23 9:14PM')
INSERT INTO Users2 VALUES('5', 'okon@yahoo.com', 'Mikey', 'mike12@gmail.com', '20-Feb-23 11:58PM')
--EXEC Update_ChildUserData 'abc@gmail.com','abc','richard@gmail.com','RichardSa'
ALTER PROCEDURE Update_ChildUserData
	@Email VARCHAR(50),
	@Name VARCHAR(50),
	@OldEmail VARCHAR(50),
	@OldName VARCHAR(50)
AS
BEGIN
	UPDATE users2 
	SET CreatedBY = @Email
	   ,Name=@Name
    WHERE CreatedBY=@OldEmail and Name=@OldName
	UPDATE users2 
	SET email = @Email
	   ,Name=@Name
    WHERE email=@OldEmail or Name=@OldName
END
Screenshot
| Id | 
email | 
Name | 
CreatedBY | 
LastLogin | 
| 1 | 
abc@gmail.com | 
abc | 
abc@gmail.com | 
14-Feb-23 8:05AM | 
| 2 | 
sammy@gmail.com | 
abc | 
abc@gmail.com | 
17-Feb-23 1:35PM | 
| 3 | 
manny@gmail.com | 
abc | 
abc@gmail.com | 
20-Feb-23 6:48PM | 
| 4 | 
mike12@gmail.com | 
Mikey | 
mike12@gmail.com | 
20-Feb-23 9:14PM | 
| 5 | 
okon@yahoo.com | 
Mikey | 
mike12@gmail.com | 
20-Feb-23 11:58PM |