Hi
Your script should update the record assuming the ID and name exist. Try a commit after the update to save the change.
UPDATE table_one
set ID=replace(ID,'&ID','&ID')
where Name='&Name';
commit;
select * from table_one; -- will show your change
Don't perform the update in one sqlplus window and try and view the results in another as the results won't show until you've issued a commit statement.
Hope it helps.
Merry Christmas
Alistair