There is no reason why they shouldn't, it's not as if you're leaving for a holiday. I know people who have left months before their official leaving date, however bear in mind the 'exigencies of the service, we need you to man a telephone while the rest go on ranges, exercises etc'.
Also, you cannot be paid by the Army at the same time as your next employer, the army will stop your wages the day your new job starts, even though you don't finish army service until after the new job starts. So speak to the accounts dept/new boss & see if your new wages can be recorded as expenses, relocation, resettlement etc, that way you get more money!
Good luck & try not to shout at the civilians on the first day, I did & the bloke burst into tears!!!!!!!!!!!!!!! He's is now 'Target of the Month'
A lad that I work with came for the interview,got a written offer of employment.He then returned to his unit (NI) gave the written offer to his boss on the Monday was discharged on the Friday started with us on the Monday.He was RAF though but it does happen.