I'd say that 2h would be the minimum comfortable connect time from T1 to T3. If you're looking at a sweating-bullets-I'm-not-sure-if-I'll-make-this-flight then the number would be a bit lower. With 2 separate PNRs you do need to check in at the CX counter so you need to be mindful of minimum check-in times. (i.e. If you don't have your BP at T-x minutes prior to boarding you might be SOL.)

I'm not sure who flies from LGA to T3 these days. I do know that you might make a 1.5h connection on 1 PNR if you take the AC flight to HKG instead of CX from T3.

BTW, since you're coming from LGA, can't you just take a CX flight from there? (Or EWR / JFK?)