Eeehhh...!! I guess there isn't much I can do in a productive way just by worrying...
Aawww! What kind of puppy???
Short walks usually better for puppies anyway (fully vaccinated now?) and UK heat doesn't even compare, but I usually take my dog out very early am and very late evening when the weather is too hot for usual daytime walks. Always bring water and a plastic travel dish with us, and be aware of hot pavements on soft, growing pawpads!
If you can drive and have air con in your car, driving to grassy or wooded area where there's shade and the ground isn't too hot, without straying too far from the car, and being especially mindful if the dog is a breed with a pushed in face and prone to overheating...
Aaww, I'm not getting another for a long time, but I'm seriously puppy broody!