OP - With the limited details you provided, it's really hard to provide any meaningful advice. As a starting point, you suggest that you were departing BHX. That is Birmingham, Alabama, USA and there would not be an visa requirements for that service. So, what were you actually booked to fly (routing, airline) and who is the TA?
As a general proposition, it is 100% your responsibility to present all required documents for travel (passport, visa or ESTA). It is also 100% your responsibility to determine which documents you need and to have them ready well before your flight.
As a second general proposition, when you do not board as scheduled, your entire ticket is cancelled. Whether it retains any value is a matter of the fare rules for the ticket. Here, it appears that you had a restricted ticket and all of the value is in taxes or the like.
If the TA is able to rebook you at little or no cost, that is a fantastic result.
Again, garbage in -- garbage out. Provide the exact details and people here can provide you with better advice.