You need to seek professional advice here. Sometimes a self-employed person can claim invalidity benefit - e.g., after a heart attack, but it'd depend on the operation carried out.
Most self-employed people'd pay into a private medical care package, and after-care'd be paid for by "top up" premiums.
If, however, your partner'd been seeing a NHS doctor about his problem before the op, and if the procedure's carried out in an NHS hospital, then I believe you'd be entitled to free after-care, but if this is the case, then I can't understand why this particular op hasn't been arranged on the NHS anyway - unless your partner's been told that he'll have quite a long wait? If it's urgent, then a NHS hospital'd have him in fairly quickly anyway.
If the op's for aesthetic (cosmetic) reasons, then I'd imagine that your partner'll have to pay for the after-care himself, as it was his choice to go privately.
Again, you need to get professional advice. I doubt you'd be able to get any other benefits to tide you over, as it could be argued that your partner should've taken out a personal insurance which'd pay him a weekly/monthly rate in the event of him being unable to work for a while.
When a person pays for private medical treatment out of their own pocket, they have to take into consideration the costs of being unable to work during the recuperation period, but as aforesaid, if the operation's urgent, I can't see why this isn't being done on the NHS. It's exactly the same procedure, and often the job's done by NHS surgeons who're earning a bit more by taking on private patients.
If it's a case of jumping the queue to avoid waiting in discomfort, then I think most people'd do it if they could, but not if they couldn't afford to live afterwards, and as I say, in an emergency, your partner'd be sent for treatment on the NHS.