As others have said, it is normal to lose any growth buds on new bare-roots, so that's not a concern. The rose has to grow hair roots before it can take up enough moisture to support growth. This may take 3 or 4 weeks. It's a good idea to mound soil over the cane stubs or spray them with anti-transpirant to reduce moisture loss. 50% manure and alfalfa is too rich a mix. I don't know whether this will have bad consequences or not, but the danger is not heat so much as excessive nitrogen that can burn hair roots and foliage, or excessive salts that can inhibit water uptake.

Your photo doesn't have the sharpest detail, but here's what came to mind: From your photo, your rose doesn't look half bad. Some of the leaves show pale yellowish mottling, which could be a sign of spider mites. To check for mites, take the plant into the sun and turn the yellowish leaves over to look for tiny specks. Rub the back of the leaf lightly with the tip of a pencil and see if any critters start moving around. If you do find mites, wash the leaves with a stream of water. Your rose looks small enough that you could cover the soil with newspaper, then hold the bush upside down under a stream of water. Wash the leaves every day for 3-4 days and the mites will be gone. Although, if they return, you'll have to repeat the treatment. How do the canes look -- green, yellow or dying back from the tips? Some extra TLC might help your rose -- liquid seaweed, Superthrive, Espoma Bio Start, Mykos, among other root and plant stimulants might help it establish a good root system. What kind of soil did you plant it in? A potting soil such as Miracle Gro or Fafard are pretty good. Soil from the garden is usually too heavy and prone to compacting.

Remember this; As long as the canes are green, it's not dead. Most David Austin roses slow down to a crawl when it is extremely hot. Somehow give it shade from noon on. Keep it moist but not soaked. Don't feed it until next spring. Right now it just wants to grow a good root system. 1st year sleep, 2nd year creep, 3rd year leap. You'll see a difference next year and the 3rd year it should go crazy.

Those pots are way too small! Roses need some room to wiggle their toes in. You need to look for pots that are at least 15 to 20 inches wide and at least that deep. Like I said find resin or plastic pots. clay is not good and will be too heavy in those sizes. Take off any of the black tips but Kate is right. The heavy brown ones that look like tree bark are good canes just older ones. You want to leave those alone. When the temps cool off get them in the bigger pots with good potting soil, keep them watered and be patient. they should recover.