How long after lcwra decision will I get payment/ back pay

Without knowing the dates of your assessment periods or what date you usually receive your money each month it’s not possible to tell you exactly when it’s paid from. 

Any money owed will be paid separately from your usual monthly money and can often take a few weeks to pay what is owed.
The above information is assuming you’re not claiming UC with a partner that already claims the LCWRA element on your claim. 

Your waiting period is June to Sept and first payment will be October 2nd. If either yourself or your partner are working and you don't currently have the work allowance, you will be owed some extra money for this. The work allowance will apply from your June/July assessment period.

Details here about work allowances. https://www.gov.uk/government/publications/universal-credit-work-allowances/universal-credit-work-allowances

The figures in the work allowance apply to the new April Increase so last year they would have been lower.