I searched Google and the results were only 嚟唔切 and not 開心都嚟唔切.
2 Answers
開心都嚟唔切 is not a set-phrase, per se, but a clause made up of different words.
If we look up the different usages of 嚟唔切 we can get an idea of what the sentence means.
嚟唔切
ABC Canto
R.V.
to be unable to arrive in time; to be too late (for something)時間比較趕,所以我嚟唔切交功課
si4 gaan3 bei2 gaau3 gon2, so2 ji5 ngo5 lai4 m4 cit3 gaau1 gung1 fo3
The time was relatively hurried, so I couldn't arrive in time to turn in my homework到老嗰陣你後悔都嚟唔切喇
dou3 lou5 go2 zan6 nei5 hau6 fui3 dou1 lai4 m4 cit3 laa3
When you become old it's too late for you to feel regret老婆走咗,恨都嚟唔切㗎喇
lou5 po4 zau2 zo2, han6 dou1 lai4 m4 cit3 gaa3 laa3
If your wife runs away, it will be too late for you to regret itSee also 嚟 lai4 唔切 m4 cit3
JP lai2 m4 cit3
not being able to make it in time
广州话方言词典
來不及。
The Mandarin equivalent would be 开心都来不及了. This equates to something like, "couldn't be happier." Sometimes it can be used sarcastically so you'd need to take a closer look at the context.
Barbie on italki addresses this usage of 來不及:
來不及 in this sentence means "you're too happy to think about anything else." Literally, you only have time to be happy and have no time for other things.
More examples:
失去了心愛的人,他傷心都來不及了(怎麼還會想再去約會呢)得到了一千萬,他開心都來不及了(怎麼還會想再繼續工作呢)
XX都嚟唔切 /XX都來不及 = excessively XX
嚟唔切 (can't make it in time) --> in a hurry --> urgent --> take priority --> ignore everything else --> excessively
開心都嚟唔切 = excessively happy; couldn't be happier
other examples of v + 唔切
走唔切 = can't get away in time
走都走唔切 = urgently getaway
走唔切(can't get away in time) --> get away in a hurry--> urgently getaway
~
做唔切 = can't keep up with the work
做都做唔切 = urgently work
做唔切(can't keep up with the work) --> work in a hurry --> urgently work