Hello Harry
I don't know if you will get an answer from someone with experience with all three rifles so I'll take a shot at a different route.
You didn't say what weight bullets you intend to use and just which model of Tikka or Winchester 70 you are asking about so this information has some guesses involved. I used the standard factory data for the Sporter version of the 70 and the Hunter version of the Tikkas.
The Model 70 has a 26" barrel so you'll get very close to the factory stated velocities for most ammo. The Tikka .300 WSM has a 24" barrel so from chronograph tests expect about 100 FPS less than factory or reloading manual data unless is state it was generated with a 24" tube. The WSMs and other short magnums seem to pay a larger velocity loss penalty for shorter barrels than the standard magnum which have more powder burning to help compensate. The Tikka 30-06 has a 22-7/16 barrel-expect 40 FPS less than the book data. If the 30-06 data was developed with a 26" barrel the loss will be closer to 80-100 FPS.
Now for the recoil. I assumed a little less than maximum powder charges and ran the weights of the rifles, bullets and powder charges through a recoil calculator. Since the most popular bullet weight for the .300s is 180 grains I used that. I pulled the expected velocity off for the shorter barrels. I used the factory stated weight and added 18 ounces for a scope and rings.
The Tikka 30-06 (7.8 pounds) with a 180 grain bullet at 2635 FPS has about 19.4 foot pounds of recoil.
The WSM (8.2 pounds) at 2800 FPS will have about 23.0 foot pounds of recoil. T
Your Model 70 (8.5 pounds) at 2880 FPS has around 25.6 foot pounds of recoil.
It is worth noting that a lot of experienced folks think that 20 foot pounds is the upper limit for most of us mere mortals to shoot without recoil becoming much more likely to cause flinching etc.
I also ran all three velocities through a ballistic calculator to compare the trajectories. They are all within an inch of each other right out to 250 yards in both elevation and windage for a 10 MPH wind. For most any applications there is darn little to pick from downrange. At that distance each of them is capable of doing anything the others can.
If you hand load, I'd stick with the Model 70 and load some softer shooting (and recoiling) loads if recoil is an issue. The .300 WM will have a bigger velocity advantage with heavier bullets.
If you choose to buy a new rifle I'd go with the 30-06. Certainly over the life of the rifle you'll save a lot of money on ammunition and you can walk into any ammunition seller in the world and find 30-06. The 30-06 is lighter, has a softer recoil and is a lot less money to feed than the magnums.
Heck, if you don't reload, save the money from the rifle and buy a set of reloading equipment.
If you have different bullet weights or specific models in mind the recoil trend will be the same but I could run the data again.
I will also note that stock shape and the total surface area of the recoil pad have a lot to do with perceived recoil. You may want to measure the height and width of your Model 70's recoil pad and then check the Tikka. If one is appreciably larger it will effect how you perceive the recoil. For example. Weatherby rifles tend to have quite large pads on their heavier recoiling guns and the comb slopes forward so that it recoils out from under the face.
Hope some of this helps.
Lance