Boxtops knows in detail how the NHS works; my answer is the generic one.
I don't believe you are entitled to demand instant release from your contract, having given the required one month notice. Some organisations insist that resigning employees using up their AL to avoid paying any accrued holiday as monies (on which you will have to pay both NI and tax). The employee can't prevent this, by demanding to work up to the last contractual day.
However, as you are currently signed off as 'sick', the employer would have no recourse to getting you to work - and they would therefore have to pay you in salary.
The problem (as I see it) is that you admit that your GP is going to sign you off as 'fit' to start the new job next week. If an employee did this to me, I'd be remarkably hacked off. If I then found out that that same employee was no longer 'sick' I'd very much want to withdraw the extra holiday pay in cash in the final salary payment (which you won't get until the monthly payslip in late March). So there is plenty of time for the NHS to uncover your little scheme.
If you insist on starting work next week in the new job, you have to be 'fit' and if you are 'fit' you are not sick as an NHS employee.
I don't see that you can play this both ways - you want cake and also eat it.