According to what I know, わけにはいかない means that someone can't do something for moral reason.
That sounds rather simplistic. 「Verb + わけにはいかない」 can be used to say that one cannot do something for other reasons as well such as laws/rules, common sense, judgement based on experience, sense of obligation, etc.
is a completely natural-sounding one that requests earnestly to see a doctor on that day (rather than on the next). Perhaps, the speaker arrived at the hospital a little too late for outpatient reception or something. It expresses the speaker's distress and strong desire, knowing that the receptionist could easily turn down the request.
As far as the phrase choice, this is in no way a 'normal' or 'neutral' way to make a request. Had the speaker come to the hospital on time and/or with everything s/he needed such as a referral letter, health insurance card, etc., s/he would have never said 「突然ですが、今日診ていただくわけにはいかないでしょうか。」.
「今日診ていただけませんか。」 (You used いただく twice, so I deleted one.)
does not imply much of the distress I discussed above. You could use the sentence, but it will not sound nearly as desperate as the one using 「わけにはいかない」.
does it change the meaning or is it just a degree of politeness?
It changes the meaning as explained above. Politeness has little to nothing to do with this. It is just how seriously and desperately you would sound regarding your desire to see a doctor.