Firstly you need to accept that it will take time to sort this out. You have not been scammed. Had you won at very high odds and were owed a million and then you were told that there was some technicality meaning the bet was invalid, then that would be a scam.
You should talk politely to the bookmaker, explain the situation fully and honestly and ask them to suggest alternative solutions. For example, if you sent a copy of the form you used to report the theft of your passport, they may accept that. Otherwise you can investigate other forms of photo id (CitizenCard.com, for example, costs £15, other low-cost ID schemes are listed here http://www.pass-scheme.org.uk/for-young-people/ ). You should be clear with yourself and with the bookmaker that, if they want photo ID, it will take a few weeks to obtain something suitable (bookmakers should be familiar with the PASS scheme, CitizenCard, etc but it's worth checking that they will accept the card and release your winnings before you apply for a card).