Some spelling errors that made this hard to understand:
great -> greet
Clint's -> clients

Now as to your specific question. You want to know good phrases to use with British clients, that are friendly and welcoming but still somewhat formal. I'm not completely sure about this. I can tell you some phrases from American restaurants, but some things that have become common in restaurants here are actually annoying. And I'm not sure if the British expect more formality than Americans do.

I would use "welcome, how are you, hello" but not "what's up" or "how you doing?" (too informal).
Checking up on tables you can say "Is everything satisfactory?" or less formal "How is everything?"
Leaving: "Thank you", "Please come back", "I hope you enjoyed your meal".

I take it you mean greet- not great?

A lot of the suggestions you put here are a bit informal for dining, especially "what's up?" I am British and I think I would find that quite rude in this context. The key is to be friendly but formal. I am a British translator and I have worked in a lot of restaurants to earn extra money whilst studying so I feel qualified to say this. Here are some suggestions:

to greet the customer on arrival: Hello/ Good afternoon/ Good evening
Asking how many people: Table for how many? Table for two? etc.
before taking order: can I get you anything to drink?
Taking order: So, What would you like?/ What can I get for you?
checking customer is satisfied: Is everything ok for you?/was everything ok for you? (also say- was everything ok for you- if you want to check whether you can take the plates away)
saying goodbye to the customer: thank you, Goodbye

It's best to stick to these simple phrases. Don't try too hard to sound like a native by using phrases like "what's up?" (which is more American anyway) rather, stick to what you know and concentrate on politeness only. British people will probably find this all the more charming because you are foreign.

"Hi, (or hello), I'm -----; I'll be your server today. Today's specials are ......; I personally recommend ...."

When you serve the food, "Is there anything else that I can get for you?"

During the meal - and not when everyone's mouth is full, "How is everything?" or "Is everything satisfactory?"

Do NOT ask, "Would you like change?" when you take their payment if they pay at the table rather than at a cashier. "Change" is money back. If you think you're too busy to return with proper change - and you Never should be because it will reduce your tip just as much as asking the "would you like change" question, you should ask "Would you like coins in your change?"

Personally, I feel it is rude to ask the customer Anything about their change: I just bring them all they have coming to them. Pressuring them or making them feel uncomfortable with either of the above questions will only reduce your tip.

Do Not address women of Any Age (particularly the older or elderly; it emasculates them) as "deary," "sweetie," "love," or the like. It's overly familiar and insulting. If you must address them by a name/title, say "miss," "madam," or "ma'am."

And, as they leave the table, say "Thank you, I trust you enjoyed your meal and my service; please come again" even if they left NO tip at all. It will speak well of you and some nonpaying member of that party may return at another time and more than make up for that missed tip.

If they are British then it's best to avoid addressing a group as "guys" unless it was an extremely casual establishment. You can say things like "Good evening", "How are you, sir/madam?" "Enjoy your meal", "Is everything all right with your meal?", "I hope you enjoyed your meal", "Thank you, please call again", "Goodbye", "Good night".

Try to avoid "See you later" -- you're not going to see them later -- use "good night" instead. And "how are you" is better than "how you doing".

PS: I think it's patrons rather than customers of a restaurant -- not clients, at any rate.