Hi OP
I guided someone through this process at Durham for a very similar course last year so I hope my experience can help you.
Firstly much as it may not seem it, it's ideal for him to have reached out now as opposed to attempt an exam again as, as mentioned above, if you get a passing grade in an exam (even if that's 42% when you'd be expecting 60+ normally) then Durham tends to stick with that. With that in mind I would suggest the best course of action is not to sit any exams (though I think that may be his decision regardless).
The first thing he needs to do is to get in contact with his department. We often found that the head of department was more useful than the general admin team, but it's worth keeping both copied in. If he's still there I think the head of Nat Sci is James Blowey, he has always been helpful for those I know who have been in contact with him. Let them know he is unwell (or if they are aware of this, that he has worsened) and will not be able to sit his exams this session and let them know he will be speaking with his GP for evidence of this and he will subject a SAC (Serious Adverse Circumstances) form by the relevant deadline. If possible he should do this tomorrow, there will almost certainly be staff in as exams start tomorrow and hopefully this would avoid any panicked emails asking why he hasn't turned up on Tuesday.
Which college is he at? College welfare (the staff, not the fellow students!) are the experts in navigating this process so I would highly recommend you or him getting in contact with them (as said by others, they will not talk to you about him without his permission to do so, but they can likely give generic advice prior to that regardless). If he wants to go and talk to someone in person about it college are an excellent place to start also. College can also write a supporting statement for a SAC application if they have been kept in the loop so they are useful people to have around.
Depending on if he has had any contact with them throughout the year another good person to contact may be his academic advisor. For the person I was helping, they were a great advocate within the department for them, but they had spoken a fair bit within the year so I appreciate if they've had little contact your DS may not want to contact them.
The next stage will be gathering evidence for the SAC form. If he has a GP registered in Durham they have an easy standard process for this (although I believe it does cost money unfortunately). It can however take a few weeks so it should be kicked off this week. He'll need a GP appointment and to ask for a green form which they will ask him to fill out with some basic info about what's been happening and what he wants them to recommend for his assessments (they were very happy to recommend deferral to a later assessment period for less serious circumstances so I am confident they would do that for your DS). It is worth considering whether the resit period (August) would be enough time as if you successfully defer to then and are then not ready you may have to go through this process a second time then. If he is registered with a home GP then I would suggest you want a similar output from them - a letter stating problems experienced, how they affected the work& caused exams unable to be sat, what solutions were tried and a recommendation for what should happen to the assessments.
The SAC form itself will have a deadline, likely in early June but there will no doubt have been an email giving the specific date or the department should know. You write a statement yourself (or you can help him write it), there is a "guide to writing a SAC form" which is useful to refer to, you tick which assessments were affected and attach the evidence collected from the GP and college if applicable (and anyone else relevant). It is more important to submit the form on time (with evidence to follow if necessary) than have all the evidence as late submissions may not be accepted, but hopefully since you are starting now then that should be enough time (and there's no advantage to submitting it earlier other than it being off your plate).
The "result" of the SAC form won't be shared til the day everyone gets their results (so sometime in the penultimate week of June this year if I have my dates right). Different departments evaluate forms separately so it would be possible to have different decisions from physics and maths but I would imagine unlikely for a situation like this.
He can also put in a second SAC form (or include it in the first) regarding any project he might have submitted recently if he feels that has been affected also. It won't adjust the mark, but it means if at a point he does sit the exams and ends up on a borderline of of a grade they may award the higher grade.
I hope that is helpful, as said by many pp it will absolutely be okay in the end, but I can appreciate it being a bit of a shock right now.