The concept of a back up offer is that the vendor can choose between one or the other. The conditional offer had just that...conditions. If the vendor accepted that first offer with the conditions then the vendor must wait until those conditions are removed and then the vendor is free to move on to another offer.
But what you are telling me is that you made an offer and it was accepted.
The basics of contract law are offer, acceptance and consideration. The latter means that some deposit or promise thereof was made to back up the offer. If you have an acceptance, the vendor has a big problem if he/she accepted two offers.
The real estate agents must be scrambling because they could be in trouble as well.
Having said that, if your offer was accepted subject to the conditional offer failing, then you remain in second place and the conditional offer prevails. It is all about the wording of each contract which of course I cannot see.
Your agent should be very concerned and should share all information. If your offer was accepted with no conditions, trouble is just around the corner because you have a law suit against the vendor and your agent.
You should ask your agent for all paperwork including the acceptance and take such to your own lawyer if you wish to proceed further.