how do you call in the UK and Ireland these american english terms?

The terms are all recognisable in the catering trade.
BUT in England:
Parking Valet is sometimes used. A valet is a personal servant. A gentleman's gentleman.
Bellman is traditionally bell boy. 'Boy' has no racist connotation in the UK. As already said, 'they are usually 'luggage porters' (´porters' may also be kitchen porters so the adjective is used).
Bus boy is not usual. They are commis waiters. (pronounced Commie)
A concierge is a concierge.

-VALET: You park your own car over here. I wouldn't consider giving my keys to anybody.

-WAITER : The same. We also have waitresses

-BELLMAN: Porter. However many small hotels don't have them.

-BUSSER,BUSBOY/BUSGIRL: Waiters and waitresses do it for themselves.

-CONCIERGE: The same but only in big touristy hotels.

We have a much more "Do it yourself" culture.

Valet - This is used more as a name for a personal manservant. There's no particular name for the person who parks your car. Or you may come across car valeting services who will clean your car both inside as well as outside.

Waiter - same word.

Bellman - this side of the Atlantic, the word is porter. (It comes from French, actually - the French word "porter", pronounced "portay", means "to carry".)

Busser - we don't have them. Just waiters and waitresses. Even in a "posh" restaurant, the waiter or waitress assigned to your table will do everything except perhaps if there is a separate wine waiter or sommelier who is a specialist and able to advise on your choice of wine.

Concierge - same word. Or if the hotel doesn't have one (many don't), ask the receptionist.

Talking of cars, if you're planning on driving, remember that we drive on the left instead of the right, and a hire car will be a complete mirror image of what you're used to except for the foot pedals, with the driver's seat on the right. You are also very likely to be offered one with manual transmission (a stick shift) unless you specifically ask for an automatic - "manuals" are far more common here. I, for one, have never driven anything else. Even if you don't plan on driving, remember that traffic is coming from the other way for safety when crossing the road. Relying on public transport and taxis works well as public transport is so good - I'm the only person in my immediate family who even HAS a driving licence. My dad never learned to drive and the idea of learning scares my mum witless.

Valet - We park our own cars in UK. wouldnt trust anyone to park my car lol.

Waiter - The same we also have Waitresses

Bellman - Porter, though rarely seen here in the UK only in very big "posh" Expensive Hotels.

Busser - Nope, The waiters/ waitresses are paid to do this without the help of others.

Concierge - Same but largely seen in bigger tourist hotels, Many smaller hotels just have a big stand at reception with alot of attraction leaflets to look through.

Valet??--You park your own car
Waiter--Same
Bellman--Porter
Busser(etc)--Not Here(UK)
Consierge--Same
If you require help and assistance on any aspect,please do not hesitate to contact me.Friendly,open,compassionate,good sense of humour.
info-at consierge-consultants.co.uk
Have a wonderful time on your vacation.Best Wishes Jim

A club I worked in did have the staff park your car. I can't remember what the term used was ( people used to take it in turns). What I do know is that if somebody checked in a high powered sports car and settled into the club for the evening, the car would have done about 50 miles or more by the time it made it as far as the car park next door. Car Valet sounds familiar, though this is often used to mean a guy in a garage who cleans the vehicles.

Waiter + Busboy:

These are terms which are common to the industry worldwide

Bellman/Concierge - don't know, as have not the experience of the hotel business. Maybe the same, though I think 'doorman' was used for 'concierge' years ago. This term now applies to the security men / bouncers instead.