Whether benefit can be backdated following a change of circumstances depends on the precise nature of the change. Tell the DWP or local authority about the change as soon as you can. For most benefits, if you tell them about the change within one month, benefit is fully backdated to the date of the change.
D&A Regs, reg 7(2)(a) ; UCPIP(D&A) Regs, Sch 1 , paras 1, 6, 12 & 15

For universal credit, a change that is favourable to you will usually take effect from the first day of the monthly assessment period (see Chapter 15(1) ) in which you told the DWP of the change.
UCPIP(D&A) Regs, Sch 1 , paras 20 & 21

For personal independence payment (PIP), disability living allowance (DLA) and attendance allowance, an increase can be paid from the day on which you first satisfy the 3- or 6-month qualifying period (see Chapters 3(4) , 4(4) and 5(3) ) if you tell the DWP about the change within one month of completing the qualifying period. If you ca