MtK has that right. Read the JSP! If I could give anyone done advice about leaving, it's take a few days to read the relevant JSP (534) cover to cover plus your single service documents relevant to resettlement. You'll find loads of nuggets they will not tell you otherwise. I have a whole of dodges ready to go.
But in summary of the JSP and again MtK is correct, you can only be paid for the last bit. If you do get paid by new company in the other leave it's up to your integrity and judgement to tell the MoD (there's is no physical impediment to the money reaching your bank account). The tax man is going to know and you'll get slapped with an emergency tax code in all probability. If your pay from your new company gets paid at the end of the month there's probably only a few weeks of moral dilemma to worry about where you actually receive monies when you technically are not supposed to.
Of course if you leave to join a new public service organisation (particularly the MoD), you will not be paid twice. They know the rules