I am not familiar with RYA, but trying to compare it looks like RYA Day Skipper would be equivalent to ASA 103 and RYA Coastal Skipper equivalent to ASA 104 or 106. Would you say that is right and do you know if you can mix them (for example get RYA Day Skipper, then use that to qualify for ASA)?
I do not believe ASA will recognize RYA to the point that they will issue you an ASA certificate. I know that RYA will not recognize ASA to give you an RYA certificate, at least not without taking a course. I just checked to see about getting RYA certificate because I have ASA, and RYA said no.
As far as chartering or renting a boat, either course will be recognized as a certificate of competence and most companies will accept ASA or RYA.
ASA is basically the US certification system RYA is English. From my understanding the rest of the world uses RYA and ASA is basically US and the Caribbean. Though the Caribbean will accept either. Other places will likely accept either, though RYA is considered to be quite a bit more rigorous.