いらっしゃいませって英語でなんて言うの？

There is no traditional restaurant welcome in the UK. Some establishments can be very informal. However, English visitors overseas to places like the US & Japan are always impressed by the politeness and level of service they receive, so perhaps we should try harder!

You could simply say 'Hello' or 'Welcome', but it is more professional to follow this with the name of the restaurant.

There are many things you could then say thereafter, such as:
-Asking how many people are in the party/how many are dining?
-You could offer to show them to a table, and if they are regulars; take them to their regular table
-Introduce yourself as their server/waiter/waitress for their stay
-If your restaurant takes table reservations, you could ask them if they have a reservation, and which name it is booked under

▪ Welcome
Welcome means to happily receive someone in the place or your home.

Example sentences:
a. Hello and welcome to my home.
b. Hi, welcome to (name of restaurant).
C. Hi, welcome to our shop.

The nature of the greeting will very much depend on the specific environment.
The first example I have given would be very good for a restaurant.
The second would be more suited to a environment such as a supermarket / convenience store.
The third might be well used in a more open environment; such as an electronics store, or a garden center.
The possibly difficult words here are:
Assistance - help, activity or attention for a specific purpose
Intrude - to be involved where you are not wanted / needed
Hesitate - wait, stop before doing something

"Welcome to... (name of establishment)" is a very common greeting.
And "Please don't hesitate to ask" is another alternative to use.
For example:
"Hi there, welcome to Samsung. Please don't hesitate to ask our staff if you would like any help." :)
And a smile is used after saying out of kindness and politeness.

All of these sentences can be used to greet people while entering a restaurant.

The traditional message from a salesman to a customer in the UK would be: "Can I help you?" but this is a little dated and predictable these days and almost always cues the response: "No thanks, I'm just looking." Salesmen shy away from closed questions these days.

If you go to a restaurant in most western countries, you will be welcomed by the staff member saying things like:-
a) Welcome
b) So glad that you could join us today
And this greeting will be followed up be asking you either
a)Do you have a reservation (If you have made a prior reservation, you will be asked your name and then taken to your table)
b) Table for how many or how many will be dining? (If you have not made a prior reservation, the waitress will then ask you about the number of people dining so that she can allocate a table with the correct number of seats for you and your dining party.