VFR transits of Class D will be given subject to 'traffic conditions and controller workload'.
I've had VFR through Luton several times (Stansted is another matter).
Having a transponder will improve yor chances, but if your position reports are accurate, your RT clear, concise and professional, then the controller is more likely to have confidence that you won't be a problem and clear you accordingly.
Best to have a plan A that assumes you won't get a clearance.