1. You can file a petition for grant of Letters of Administration (with or without Will annexed, depending on whether your mother left behind any Will).
2. In the meanwhile, since your brother is illegally attempting to sell your mother's properties, you can issue a public notice in newspapers informing the public not to deal with your brother in any property transaction since you being the legal heir, have also a claim on that property.
3. You can also file a suit against your brother claiming the relief of injunction i.e. restricting your brother against selling any of your mother's properties and for a declaration that you are entitled to her property as a legal heir.
4. If your brother claims that your late mother left behind any Will, then the Will alone will not suffice for transferring the amounts lying to the credit of your mother's bank account, without it having been proved before a Court of law by filing a probate Petition (in case the Will has an executor) or a petition for grant of Letters of Administration (if no executor is appointed in the Will).
5. Simultaneously you can also lodge a police complaint against your brother for fraudulently and illegally attempting to sell your mother's properties without your consent. This is just to deter and pressurise your brother. Expect no positive outcome from the police complain since the police will say that its a civil dispute. But its always better to attack from all sides and shake the opponent.
6. Coming to the time involved in above proceedings, then as to filing of Probate or LA petition, it being departmental, would take around 6 months, provided there is no challenge by your brother. If your brother challenges the grant of Probate or LA, then such petition will be converted to a regular suit.
7. Nobody can predict how long a suit will take to get resolved. Lawyers should not be expected to be astrologers since a suit may take many different twist and turns. Generally you can bear in mind that once a suit is filed, it will take atleast 5 years for resolution. This is rather a conservative estimate. Some suits take several years for adjudication.
8. Next are the charges. For any testamentary petition like probate or LA, you will have to bear fixed court fees plus clerkage plus lawyer fees. For suit the fees depend on the facts of the case.
9. I would suggest that for the present, you issue a public notice and also a legal notice to your brother and see how he responds.
10. Since its a family matter, I will advise you to amicably settle the matter with your brother. This will ensure that your mother's soul is not pained by the fact that her children are fighting over her property.