Wow! I'm sorry for being so wordy about this topic!
But something came back to me as I was thinking further about your tire question & my own Hubley kit build from 40-plus years ago.
If your kit's tires are still fairly stiff or seem too tight of a fit on the model's wheels even after warming them up, then applying a small amount of soapy water around the tire's inside diameter and around the wheel rim should help them press into place at least a bit more easily.
When you think about it, it's essentially a scaled-down version of how actual car & truck tires are installed on rims. Instead of tire soap, though, I'd say one drop of Dawn or similar dishwashing soap mixed in a cup of water would be about right.
Going further, if things still seem so tight that you're worried about breaking something when you press the tires on, then try rolling up a small piece of fine wet/dry sandpaper & lightly sanding the tire's inside diameter. This should be the ultimate fix. Roll the paper up lightly so it's got a slightly larger OD than the tire's ID. Push the paper through the tire, then turn the tire around while holding the sandpaper still, like winding a clock. This should remove just a touch of tire material evenly all around its ID without making the tire out-of-round. Two or three tries at the most should do it.
I realize this an awful lot of fuss for softening hard old tires & getting them to fit, but you sure don't want to crush any of those virtually irreplaceable wheels.
Of course, you can always make your own spoked wheels. Seriously. Google Gerald Wingrove, MBE & look at some of his car modeling work. It's truly mindblowing to see how gifted this man is. He makes virtually every part of his models from scratch & even scratchbuilds many of the tools he uses to make the parts with! He's made countless amazing car replicas for various museums & private collections around the world. He is also a master ship modeler, has written several outstanding modeling books & is an inspiration to modelers of any skill level. He is British & has been awarded the Most Excellent Order of the British Empire for his contributions to the arts and sciences.
Best of luck & I hope this will help you. But if all you truly wanted to know was how to make hard old tires softer, I apologize for all the unwanted verbiage & unsolicited advice. I'm sure this isn't your first kit building experience, but I don't get very many chances to talk about building models now that I'm retired. If I say words like "kit" or "model" to my wife she hides under the bed. But it's never the same bed twice which seems quite unfair. Just kidding, doll!.
I still recommend checking out Mr. Wingrove's work, though. He's far more interesting than I am! But I wonder if his wife hides under beds, too?