Everybody needs a martial weapon proficiency

The shield bash rules say this (emphasis mine):

You can bash an opponent with a light shield or heavy shield, using it as an off-hand weapon. See Table: Weapons for the damage dealt by a shield bash. Used this way, a shield is a martial bludgeoning weapon. For the purpose of penalties on attack rolls, treat a heavy shield as a one-handed weapon and a light shield as a light weapon. If you use your shield as a weapon, you lose its AC bonus until your next action (usually until the next round). An enhancement bonus on a shield does not improve the effectiveness of a shield bash made with it, but the shield can be made into a magic weapon in its own right

Shield proficiency says this:

Benefit You can use a shield and take only the standard penalties.

Normal When you are using a shield with which you are not proficient, you take the shield’s armor check penalty on attack rolls and on all skill checks that involve moving, including Ride checks.

Shield proficiency doesn't give you any kind of weapon proficiency with shields. It gives you the ability to use the shield and not take the armor check penalty. In order to not take a penalty for using a shield bash, you need that martial weapon proficiency. Classes that give you all martial weapons take care of that pretty easily, like a level of Fighter (or Radiant Servant of Pelor if you're a Cleric).

What about "Simple Weapons Only"?

The other classes you mentioned actually have a list of what they're proficient in, the Cleric's "simple weapons only" is just shorthand for a list that has every single simple weapon on it. It doesn't mean that Shield Proficiency works differently for Clerics than it does for everybody else.

It's also worth noting that the Cleric says "all simple weapons", not "only simple weapons". It's not a special rule or exclusion, it's just a shorter way of saying it than listing every simple weapon.

Pathfinder - The Same

From looking around, this appears to be the same in Pathfinder. The shield bash rules are basically the same, the shield proficiency rules are worded a bit differently but don't affect this, and James Jacobs (Creative Director at Paizo) agrees.

In fact, the Paizo FAQ states clearly that shield proficiency != weapon proficiency with shields.

Yes, you can

PH170, the feat in question, says "[y]ou gain the following benefits while you are wielding a shield". The feat says nothing about wielding a shield you are proficient in—otherwise it would be on the prerequisite line—just that you have to wield a shield.

Of course, if you're not proficient with the shield, many of the benefits are moot.

While adding your shield's bonus to Dex saves and gaining the ability to take no damage from a successful Dex save those saves are going to be harder to succeed at with disadvantage.

Per PH144:

Armor Proficiency. Anyone can put on a suit of armor or strap a shield to an arm. Only those proficient in the armor's use know how to wear it effectively, however. Your class gives you proficiency with certain types of armor. If you wear armor that you lack proficiency with, you have disadvantage on any ability check, saving throw, or attack roll that involves Strength or Dexterity, and you can't cast spells.

Therefore, not being proficient with a shield grants disadvantage on Dex saves—the same saves the shield master feat is meant to improve.