Whilst it's a technical point, when you said that all of your convictions arose from the same situation, I assumed you meant from the same individual incident. For example, you could have been convicted on two counts of common assault if you punched two different people in a fight. In fact you have been convicted of numerous offences over an 11 month period, one of which resulted in a custodial sentence. On the face of it I would have thought that you would struggle to be admitted either by the SRA or the BSB.
That said, I have no experience of this situation either, and I would agree with J-SP that you need to contact the regulator(s) concerned and get information directly from them. You should be aware that if your ambition is to become a barrister you will also need to be admitted to one of the Inns of Court, and criminal convictions can cause you difficulties in that respect. You should therefore contact one or more of the Inns as well as the BSB if you want to be a barrister. I don't know if it a person would realistically be rejected from the Inns if they have been admitted by the BSB, but you do need to consider this issue from the perspective of the Inns as well as the BSB if that's the route you want to go down.
As a final point, if you get a negative response from either or both of the regulatory bodies, I would ask whether it would make a difference if the offences were more historical in nature. That won't help you right now, but with the passage of time (and, presumably, a long stretch where you have demonstrably improved your character) it may be that the situation changes somewhat, meaning that if it really is your dream to work in the legal sector, it may be something that you can pursue further down the line.