“Veuillez” is the second person plural conjugation¹ of “vouloir” in the impératif présent. “Allez à l’arrière du bus” in french is a direct order; “Veuillez aller à l’arrière du bus” (litt. “Want to go to the back of the bus”) is softer. By consequence, it is indeed only used when you ask someone to do something. In English, you should indeed translate it by “please”.

However, it is perfectly compatible with “s’il vous plaît”:

Veuillez aller à l’arrière de l’autobus, s’il vous plaît.
Veuillez s’il vous plaît aller à l’arrière de l’autobus.

are both perfectly correct and should be considered more appropriate than the no-s’il-vous-plaît counterpart.

S’il vous plaît, allez à l’arrière de l’autobus.

is correct but more of a plea. (Please go to the back of the bus ?!)

Allez à l’arrière du bus, s’il vous plaît.

is much stronger than “veuillez”, more akin to what a teacher would say its student than to what you would say to a stranger.

1. Actually, the impératif présent of vouloir is only used in the 2nd person plural, never in the 2nd person singular or 1st person plural.

I guess I’d translate it with “would you (please)…”
As stated before (and same as in english) it is a softer imperative: Instead of saying “please move to the end of the bus” you’d say “would you please move to the end of the bus”

IN humble opinion, it is better: Veuillez + infinitif = Would you please + infinitive.

S’il vous plait = Please.

S’il vous plait is used for requests, but veuillez is used for demands.

I would think the (American) English equivalent would be “Do you want to….” or, in slang “Ya wanna….” For example, someone is coming toward the house struggling with several bags of groceries and says to me “Ya wanna take one of these?” That actually means “Would you please take one of these?” There’s nothing rude or impolite about it. (Unless someone is being a bit passive agressive – which certainly happens.

It seems that “Kindly” is a good English replacement. As in “Kindly” remove your shoes before entering. Or “Please” remove your shoes before entering.

It seems the aim is to be courteous when giving instructions or commands.

So “Please” is a good translation but “Kindly” will probably help English speakers understand the subtle but important difference.