Running after hip surgery

I wouldn't dream of it.

I have a friend who does a lot of horse-riding which she was anxious about post surgery.

I can't imagine why you can't jog - I'd start gently and choose the best footwear.

I am having a replacement and was told that newer hips are constructed of different materials - I think high-impact activity was a no-no with the old ones.

If your surgeon is confident then I would be too.
It seems to be to do with current stability and muscle tone, balance etc.
Your surgeon will have seen you.
Trust them.

Following my complex joint surgery I found that the physiotherapist had more time to discuss my activity going forward that the surgeon did.
Apparently, the physiotherapist I was assigned to was actually observing in theatres when I had my operation - she is very highly qualified and has decades of NHS service, she got the chance to observe and took it. Sheer luck that I was assigned to her.

NotSpaghetti : my surgeon is one of the team who developed the technology that got Andy Murray back on court. He's also a national lead in hip surgery and has referrals from all over Europe. He seems pretty confident he can get me back running shorter distances (I was never planning to do more than 10K, but 5K with the dog will be just fine) and is using ceramic / polyurethane technology which is better for active younger people. It may not last quite as long, but by the time it needs replacing I'll be nearly 70 and most likely slowing down by then anyway.

I'm still scared of the future though as you never can tell. I'm 20yy post cancer and still fighting!

I'm still scared of the future ...this is the real issue.
So am I after a major incident in 2022 so I do know how scrambled things feel sometimes. The "loss" of something is difficult to bear but after surgery you will no doubt be better than you are today.

You know you are in great hands.
It's easy to say but please try to relax a bit about it - I'm trying to accept things I can't change and work a bit on things I can.

I know someone who was a keen tennis player and had both hips replaced in his 60s. He was able to resume playing tennis and also his other hobby of hill walking - not the gentle kind, but serious, long distance hikes.

Yes... I have had x2 hips and a knee replaced, and yes I run on a treadmill, not street. I go to the gym at least twice a week.
I did leave it for about 3-4 months, took it easy, then slowly built it up.