So, the MySQL (and Drizzle) ALTER TABLE syntax allows you to easily change the default value of a column. For example:

CREATE TABLE t1 (answer int);
ALTER TABLE t1 ALTER answer SET DEFAULT 42;

So, you create a TIMESTAMP column and forgot to set the default value to CURRENT_TIMESTAMP. Easy, just ALTER TABLE:

create table t1 (a timestamp);
alter table t1 alter a set default CURRENT_TIMESTAMP;

(This is left as another exercise for the reader as to what this will do – again, maybe not what you expect)