Please don't hate me for how stereotypical this is but I figured it had to be done. Time to try remembering as many countries as possible for this.

Times they fall asleep-
America- between 2 and 3 in the morning

England- 10:00 pm sharp (22:00)

France- 2:00 am

Italy- varies. He can fall asleep at 9:00 pm (21:00) or 3:00 am depending on where he's sleeping and who he's sleeping with.

Romano- anywhere between 11:00 pm (23:00) and 3:00 am

Germany- 11:00 pm (23:00) every night. On the dot. No matter the circumstances.

Spain- 2:00 am

China- 10:00 pm (22:00)

Japan- 9:00 pm (21:00)

Prussia- often refuses to sleep. When he does sleep, he falls asleep between 3 and 6 in the morning and doesn't wake up until about 10 in the morning.

Canada- 10:00 pm (22:00)

Russia- 1 in the morning

Estonia- 11:00 pm

Latvia- 1 am

Lithuania- 8:00 pm

Poland- 2:00 am

Finland- around midnight

Sweden- doesn't fall asleep until he is completely sure that Finland is asleep

Denmark- 1:00 am

Norway- often either 9:00 pm or 2:00 am, it just varies

Iceland- 1:00 am although he doesn't wake up till 11:00 am

Belarus- 11:00 pm

Ukraine- 9:00 pm

Hungary- 11:00 pm

Austria- 9:47 pm (very specific, go figure, Austria)

Scotland- 2:00 am

Bulgaria- 11:00 pm

Romania- 1:00 am

Australia- midnight

Cuba- 11:30 pm

Belgium- 9:30 pm

The Netherlands- 10:00 pm

Liechtenstein- 9:00 pm

Switzerland- 10:00 pm

Greece- He's always sleeping, I don't think there can be a set time.

Turkey- 11:45 pm

Egypt- 10:00 pm

Seychelles- 11:00 pm

South Korea- 1 am

Thailand- 10:40 pm

Taiwan- 9:00 pm

India- 9:30 pm

Moldova- 10:45 pm

Cyprus- 9:30 pm

Monaco- 10:00 pm

Ladonia- 12:30 am

Sealand- 1:00 am

Kugelmugel- 11:00 pm

Hong Kong- 2:00 am

Hutt River- 10:00 pm

Molossia- 1:00 am

Niko Niko- 10:30 pm

Wy- 10:00 pm

Seborga- 1:00 am

TRNC- 9:30 pm

Macau- 10:30 pm

Vietnam- 9:00 pm

New Zealand- midnight

I believe I got every country. Whew that wasn't easy to write or remember each country.