"Belated happy birthday" or "happy belated birthday"?

I would favour the first, because a wish for one to have a happy birthday is possible, but a belated birthday is not, though the second suggests that you are wishing them happiness on their belated birthday.

A belated birthday celebration is of course perfectly possible, but I'd just say "happy birthday" in such a case.

Well belated means behind date or late.

From Oxford English Dictionary:

belated, adj.

1. Detained beyond the usual time, coming or staying too late; out of date, behind date.

So, if you're wishing someone a Happy Birthday which is late, you would use:

Belated Happy Birthday

"Belated", of course, refers to something that has been delayed. From Merriam-Webster:

be·lat·ed adjective bi-ˈlā-təd, bē- 1: delayed beyond the usual time 2: existing or appearing past the normal or proper time

So, to wish someone after their actual birthday, the best phrase in my opinion would be "Belated birthday wishes", because it's just your wishes that got delayed.

Among the two options yo've given, "Belated Happy Birthday" is more common in my part of the world than the other.

A ‘Happy Birthday’ would mean- ‘May the day of your birth(current day), happy for you’. So wishing someone delayed, say after 5 days- ‘May your birthday that occurred 5 days ago, should have been happy’ ?? umm.. It does not make any sense. Delayed birthday wishes make no sense other than realizing our mistake. Your feelings are not outdated, anytime you wish, it’s fresh. So I feel, ‘Happy’ should be the first word whenever you wish. The day of birth has gone so ‘belated birthday’ So ‘Happy Belated Birthday.

Personally, I've used both versions. Nevertheless:

• Belated birthday is nonsense, since the anniversary is the anniversary, and cannot be postponed even if the celebrations are.
• Belated happy birthday, strictly, is also nonsense because the birthday has already gone and may or may not have been happy.

Therefore I would suggest something along the lines of:

• Belated birthday wishes (as suggested in another answer)
• Hope you had a Happy Birthday. Sorry I missed it / Sorry I'm late.

A "birthdate" is always a specific date. It cannot be delayed like a dentist appointment. Just because you forgot or missed sending your greetings prior to and up to the actual date it has no bearing on the birth date. Your greetings are late or belated. Try to diagram the sentence "I wish you a happy belated birthday."

The perception that there is something paradoxical or nonsensical about belated happy birthday and its variants is due to taking happy birthday itself at face value, that is, as an expression of a wish that the addressee's birthday (one particular day) be happy. However, when people say happy birthday, their wishes are usually not specifically focused on that day. If they were, it would make much more sense to express such wishes in the morning, rather than the evening, when most of the day is already gone, but the latter is, in fact, far more common. The expression happy birthday, instead conveys vague wishes that the addressee's life generally go well, combined with the recognition of the birthday. If what is packed into happy birthday were made fully explicit, it would amount to something like: 'Noting that today is your birthday, and appreciating the significance of that fact, I wish you that your life go well in the future'.

There is nothing paradoxical about expressing such wishes with some delay. What is conveyed by a belated happy birthday message is something like: 'Noting that your birthday was a few days ago, and appreciating the significance of that fact, I wish you that your life go well in the future'.

Now, there may still be something syntactically puzzling about belated happy birthday in that belated in it seems to qualify birthday and it is unclear what that would mean (relative to what is the birthday delayed?). That puzzle can be resolved if we think of happy birthday analogously to thank you. It is generally accepted that the latter phrase can be used not only as an exclamation actually expressing thanks, but also as a noun phrase standing for something that expresses thanks. If somebody were to write 'Here is my belated thank you for . . .', we wouldn't have any problem understanding that thank you is used in the latter way, and that belated qualifies thank you as a whole. Similarly, when we receive a card saying belated happy birthday, we should think of belated as qualifying the whole of happy birthday, rather than just birthday.