Depends where and how did you apply?
1. Did you cross illegally at Roxham rd and make a claim at the refugee centre or be sent to Montreal to make a claim?
2. Did cross legally and you make an appointment for refugee at the Montreal IRCC office and fill out a basis claim and submit fingerprints ?
3. Did you cross legally, told Montreal IRCC you intended to make your claim, but never showed up for your appointment and never submitted fingerprints ( thinking this is your case if you still had your passport)
If #3 is your situation, you technically never made a refugee claim, it won’t show up in any info sharing. There will be some GCMS info you tried to make a claim and it might be a little harder to get a Canadian visa.
If #1 or #2 applies then the UK or Australia can get that information from Canada since they are part of the “five eyes” program, but only if they think to make the request from Canada.
If 1 or 2 apply did you confirm your departure before leaving? If you did not and 30 days go by, you will be deemed deported and can never come back without an authorization to return. If you did confirm departure, you can come back at any time as long as you can satisfy an officer you will go home.
You’re not banned from US travel though. Just have to satisfy them that you won’t overstay.