What to do when creeped out by Charlie Chaplin on Westminster Bridge?

1. Is it safe to shout for the police?

Always. However, from what you posted in the question it doesn't seem like these people are doing anything illegal to you. They're just being annoying, which is not a crime. Arguably scaring people might be 'breach of the peace' but I suspect the police have much better things to do. Of course if you see something definitely illegal then inform the police. And from the link in the comments and below it's pretty clear that there is illegal activity going on and the police know about it already.

2. Is it preferable and safe to immediately dial 999?

NO. 999 is an emergency number and should only be used for emergency. Unless Chaplin is stabbing someone or something like that your better bet is to contact the nearest police officer or contact the local police (the met.police.uk, as pnuts posts in the comments will have contact details). You may not know this but you can also dial 101 which is the police non-emergency contact number.

3. Or should an encounter be dealt with patience and cunning?

It should be dealt with by ignoring them and reporting harassment and criminal behavior to the police. You can try saying 'No' or 'Please move' in a firm voice if they're in your way but interaction may encourage them. Ignore them, don't act like a tourist, they'll leave you mostly alone.

4. Why don't the authorities assign more police officers on that bridge, which is right under the nose of the Houses of the Parliament?

Well the link posted above (and I post another below) says they are taking action, but my guess is it's difficult to make a prosecution stick, they may just succeed in moving people somewhere else and taking their stuff. The police have limited resources, they probably just shake the place down every so often rather than keep a permanent presence (because obviously Chaplin is going to behave if the police are right there). Perhaps more people need to complain.

5. Do these characters come out only after dusk?

I don't know. It would make sense they'd try and hit the key tourist traffic times, but also they may want to hit times with lower police presence. By not being there constantly it makes it more difficult for the police to monitor them.

6. What do these characters want?

(Warning, partially informed guesswork and speculation ahead)

I hesitate to ascribe sinister motives, but the evidence points to the fact that they're up to no good.

You mention yourself the photo scam, there are various other scams they could be running (gaming, for example, as a some sites say). The photo-bombing is probably just a pretext to get money ("Hey, you pay me for my photo" or "If you give me some cash I'll stay out your photo"). But they probably only do that if people interact with them first.

The whole thing is likely just to get people into conversation for in order to enact the proper scam (either begging, or 'fancy a quick card game'), it also filters out the locals that don't interact with them -- they know there's little point in trying to scam them.

It's also not impossible that they're serving as distractions for a pickpocket ring on the bridge. They'll target/scare/photo-bomb more people than are actually robbed so as to not make it completely obvious.

Alternatively they're trying to provoke a reaction, just either for the fun of it or so they can escalate it to the point where people may be inclined to give them money to get out of the situation.

NOTE: Edited answer in response to a change in the question, also clarified my answer to one to make it clear that while there is illegal activity going on (as evidenced by the press releases in the links) nothing mentioned in the question was clearly illegal (although it's arguable, and I am not a lawyer/policeman).