The correct syntax should be something like that, if I've understand well what you need:
CREATE CONSTRAINT ON (a:A) ASSERT a.reg_uid IS UNIQUE;
CREATE CONSTRAINT ON (b:B) ASSERT b.reg_uid IS UNIQUE;
CREATE CONSTRAINT ON (c:C) ASSERT c.reg_uid IS UNIQUE;
This create constraint without a name, but you can create named constraint
CREATE CONSTRAINT uniqueA ON (a:A) ASSERT a.reg_uid IS UNIQUE;
CREATE CONSTRAINT uniqueB ON (b:B) ASSERT b.reg_uid IS UNIQUE;
CREATE CONSTRAINT uniqueC ON (c:C) ASSERT c.reg_uid IS UNIQUE;
If you're using Enterprise Edition, maybe you want to make reg_uid a NODE KEY, so that a node of this type could not be created without it. This is the way for that:
CREATE CONSTRAINT nodeA ON (a:A) ASSERT a.reg_uid IS NODE KEY;
CREATE CONSTRAINT nodeB ON (b:B) ASSERT b.reg_uid IS NODE KEY;
CREATE CONSTRAINT nodeC ON (c:C) ASSERT c.reg_uid IS NODE KEY;
If you want to drop them, this is the way:
DROP CONSTRAINT uniqueA;
DROP CONSTRAINT uniqueB;
DROP CONSTRAINT uniqueC;
DROP CONSTRAINT nodeA;
DROP CONSTRAINT nodeB;
DROP CONSTRAINT nodeC;