To mostly reiterate what's been said, if you get rid of the oil crisis the economics improve enough that Concorde B is very likely to happen (solving the aircraft's range though not economic problems) and making it reasonably plausible for an American SST to appear. That said, the economics are still pretty awful when compared to conventional wide bodies, and more so against large twins. That said, if you delay the oil price spike long enough I expect you can generate a meaningful and globe spanning niche for SST, though not replace most conventional aircraft as the OP asked for.
My best guess is that without the oil crisis and aiming for maximum SST penetration you get Concorde entering service much as OTL, but with significantly more airline enthusiasm. I'd guess that well over a hundred would be sold between the A and B models, and that with strong Concorde sales, still reasonable earning potential with more 60's like oil prices and proven customer demand the American program goes ahead, and enters service around 1976 (could be later, although I get the impression that thing were starting to come together around the time of cancellation). In any case, Concorde would be quickly outmoded and ends up being a successful type, but still questionable financially.
Whether the Concordes are replaced quickly or not has a lot to do with how well the 2707 or whatver the American aircraft is called actually lives up to it projected performance. My best guess is that it will end up with reduced payload and increased weight, fuel burn and operating costs and while enough of a financial improvement that few airlines are going to keep ordering Concordes its not going to demand their immediate replacement either (I'd also guess that Concorde could well retain a military and civilian business jet like market). Between the two aircraft I would expect major over water intercontinental destinations from Europe to have SST service, as well as most major trans-pacific routes, but that penetration into other markets is at best limited.
The real butterflies start at this point in terms of where Airbus goes with a successful Concorde program to build on. My best guess is that the A300 goes ahead much as OTL, but I wonder if the transition to it being an ETOPS aircraft ever happens. With most intercontinental destinations from Europe served supersonically a widebody SST to compete with the American aircraft seems like something Airbus is going to want quite badly, but ultimately I suspect given the large demand and lack of overlap with the supersonic market the A320 would go ahead much as OTL, if anything the SST market might accelerate the narrow body project, eliminating the demand for the A310. Certainly the A330/A340 program seems unlikely to go ahead as OTL, Airbus would very likely go ahead with a next gen SST instead.
I think the 767/757 would happen much as OTL (bearing in mind the original mission of the aircraft as an A300 for the Americas and a 737 replacement), but delayed a few years by the financial and technical impacts of the SST program, but into the 90s and 2000s things get less clear. I could easily imagine reduced sales leading to a 747 modernization and earlier and better marketed 767-400 replacing the 777 and major upgrades (at the same kind of scale as the upgrades the 737 between the classics and NGs) to the 767 killing the 787 (ETOPS would exists by this point, but the SSTs will reduce the size of the market, which would IMO reduce the market for a very large twin like the 777). I'd guess that in the same time frame (or a few years sooner with the reduced number of all new airframes being produced) as OTL's 787 Boeing would build the Sonic Cruiser as a true SST with much improved performance over the 2707.
My guess is that in all this the big losers are actually going to be McDonnell Douglas and Lockheed, for whom the trijets will become unmitigated disasters (arguably not much of a change for the L1011). Early sales would probably come through much as OTL but follow-ups would be very limited if the SSTs find their niche IMO, and the much reduced number of aircraft would mostly be seen in American domestic service and with various charter operators in all economy configurations. McD would certainly TRY to produce an SST in the 70s and 80s but I doubt that they would have the financial ability to do so, and I suspect that the company would be bought by Boeing much as OTL, but having never pursued the MD-11 instead losing what financial capacity the company had left under the weight of a serious attempt to build an SST to compete with the 2707 in mid 90s (that said, if the company instead worked to turn the DC-9 airframe into the first of the smaller RJs they might do much better than OTL, but that would require a pretty big shift in terms of the company's perception of its place in the market).
In broad terms I would expect that todays industry would probably leave SSTs in something of the same position as the smaller regional jets... Financially undesirable in most cases, but for many a cost of staying competitive, and one that is already sunk on the capital side. SST service is probably a well established business oriented product on the Pacific, North Atlantic and between Europe, the Middle East, Southern and Eastern Asia and Australia. I doubt that supersonic overlflight of populated areas ever does happen between the noise, damage potential, persistent feelings that the SSTs are for the rich and the environmental problems associated with the aircraft, but the odds of getting permissions for central Australia and the Canadian north seem decent. Russian overflight for eastern Asia to Europe could conceivably happen in the post Soviet world, but I think by the time that would become a realistic political possibility the economics would make airlines less than eager.
Perhaps the most interesting butterflies here, other than some very cool airplanes being in service are where this leaves Airbus. While I don't think the consortium will disappear, this TL suggests to me that they may remain something of a regional supplier, continuing to cater to Europe first and formost with limited penetration in the rest of the world. While a long range A300 would seem likely eventually, I doubt that it would happen until the 767 creates the ETOPS market if the SSTs are seen as the true future of long distance travel through the 70s. A major upgrade might happen in the same general era as teh A330 was developed to keep the aircraft generally competitive with the 767, but an Airbus competing against 767NGs with A300 NGs is going to pretty strongly emphasize the age difference in the underlying airframes (though the A300 is certainly a better cargo aircraft the the 767 OTl, and this could have an effect).
All in all, this makes the A320 Airbus' big chance to break out globally, and while things could stay OTL if you assume the A320 is OTL's aircraft I wonder if that would be the case. If the lack of an A310 program were to move the program forward a couple of years I could imagine a less advanced aircraft more tailored to replacing existing European designs emerging in the mid 80s. While this would be by no means a bad aircraft it would not necessarily be such a 737 Classic/727 killer as in OTL. I doubt that European sales would be much affected, but with less of a clear benefit I wonder if it ever would break in to North America (although a lot depends on what Boeing does with the 757 as well, if they start pushing it when airlines really want an improved 737 we might still see A320s in the Americas).
tl;dr You can make SSTs common, but you aren't going to replace conventional aircraft. The resulting industry is different IMO, but defintiley recognizable, and while the butterflies exists and are interesting they also don't seem particularly large outside the immediate corporate structure of aviation and related industries.
PS:
Someone asked about low mach designs, and while I agree that there aren't a whole lot out there the proposed
Aerion Supersonic Business Jet would fit the profile.
