Hi Nino,
Just in short, there is no right answer to question „Which is the best programming language to model this scenario“. Some may say Node.js, Ruby, Python ... or any other language, but my recommendation is to
stick with the programming language you know best – .NET or Java (as you wrote you are familiar with the desktop apps).
Use
HTML5 for client web app plus one of the many
JavaScript frameworks for building dynamic web apps. I dn’t know how the Bingo game works, but you should be OK with basic styled tables. Maybe use some Canvas or SVG for advanced graphics, but it all depends on your needs.
Because your web app will need to communicate in real time, I would suggest you to use
WebSockets. See the
caniuse.com for support in mobile browsers. There is also many JS wrappers, that will try to use other technologies as a fallback if WebSockets are not supported by client browser.
Use
existing libraries and don’t try to reinvent the weel!
Also be aware of
real environment – your office with small amount of clients on one Wifi access point is not the same as 50 people on one sh@#!y Wifi.
I do not mean to offend you, but from your question it looks that it will be very challenging for you to successfully complete this project.
Hope this helps and good luck.