I am not 100% sure. When you apply to the Law Society, they do have a bit on the form where you explain your convictions (if you have any) which implies that you can, potentially, be a lawyer having been convicted of a crime. Firms also include a section in their application forms for you to disclose your criminal convictions, which again implies you can.
However, I guess whether the Law Society decides to accept you or not (and you can't be a practising solicitor without Law Society membership) depends very much on the types of crimes committed, how old you were etc. I guess that they may be a little stricter on crimes that involved deception/fraud and so on, than for example, a driving offence.
If you have a criminal record, the first issue is whether the conviction is spent. You still have to declare it, but it is far more likely to be overlooked.
You need permission from both the Law Society or the Bar Council to be admitted/called. Whether you get it or not will depend on what you did, whether you pleaded guilty and your progress since. I wouldn't commit to paying the fees for LPC or BVC until you knew what the answer would be. A failure to make a comprehensive and accurate declaration will, certainly as far as the Bar Council is concerned, lead to striking off. I cannot imagine the Law Society would be any different.
I ought to have said that every case is considered on its merits and there is no absolute guarantee that you will not be allowed to practice. But dishonesty is worse, in this context, than violence, which is usually worse than public order. Get some references that say the people writing them know exactly what you did and can say you would not do it again, ever. And not your Priest/Vicar/Rabbi/Imam - those guys are paid to be nice about people!
AFAIK you do not have to declare cautions. A caution is an admission of guilt however the Law Society form only asks about convictions in front of a court.
Cautions are classed as instantly spent and are wiped from the PNC after five years. The only people who can legally ask you to declare them are the police, security services and educational establishments. See the Rehabiitation of Offenders Act.
In any event I know people who who have been cautioned and later joined the police, I even know of a University member of staff who has been done for GBH, albeit 10 years ago.
I may be wrong but I think that there is no legal requirement to tell anyone else about them. Consequently I do not think that you have to tell the Law Society or Bar Council. But I may be wrong!