An AONE4 (First Class OneWorld Explorer, 4 Continents), starting in Tel Aviv is US$10,000 (approx £6200) + taxes. You won't get clobbered too badly for fuel surcharges if you get it ticketed by AA. You could do most of your desired itinerary in First, but Fiji is a no-no on any OneWorld Explorer as no OW carrier flies there (on their own metal). If you really must do Fiji, there is, however, the OneWorld Global Explorer fare. This allows use of Air Pacific, and other carriers such as Alaskan Airlines, and is the same price as the 4 continent OW Explorer. It is, however, limited to a max. of 34,000 miles whereas the 'plain' OW Explorer has no stated limit. (I did about 66,000 miles on my AONE5 a couple of years ago.)

Obviously, to kick off in TLV you have to get there from the UK, but Sqezy have flights from Luton, I believe. You don't even need to end in TLV, as anywhere in the Middle East is OK. You could end in Oman, which, IMHO, is far superior to Dubai, and use some of your earned miles to get back to the UK.

As many flights these days don't offer F, your itinerary could cost around £4500 + taxes (+ getting to TLV) if you were to do it all in Business class. Personally, I think paying around 35% more, to get F where available, is worth it, as a trip like this is something quite exceptional, but that is very much a matter of personal preference.

An example xGLOB34 itinerary is: TLV-LHR-HKG-BKK-SIN-SYD-NAN-HNL-LAX-SFO-LHR-MCT, which is a bit over 31,000 miles. You could also add in a trip to APW, in Western Samoa for next to nothing or do a side trip somewhere along the line, say to AKL and MEL. Side trips usually don't offer F, but the main itin. could all be in F, I think, except for the BKK-SIN & SYD-NAN_HNL portions.

Hope this helps your thinking..