if you are on 2 seperate tickets, then you are at the mercy of the airline concerned. Strictly speaking, its a new ticket, although this is rare, options I have seen range from being put on the next flight free of charge, or paying a small surcharge to do so (if the rules of the yicket permit changes).
If you are on 1 ticket, you are fine, the airline who the ticket has been issued on have the responsability to get you from start to finish
My advise - be very,very nice and polite to the ground staff