I agree with W8PRR. Most radios are designed with a 50 Ohm output in mind. So, yes, if the "only" thing you swap out is the radio, there should be very little difference in the SWR measurement.
SWR, as measured from the radio end, is a measurement that indicates the impedance match, or mismatch, between the antenna and the feedline. A much better idea of the antenna impedance can be had by measuring directly at the antenna. This removes the coax losses from the reading. But this is not always convenient or possible.
Measuring SWR from the radio end can hide problems with the antenna. For example, if you measure SWR from the radio end, and then upgrade your coax from say RG-58 to RG-8. It is very likely that your SWR will increase. The high losses in the lower grade cable, RG-58 in this example, may let you see a low SWR than really exists. But when you reduce the cable losses, by upgrading the coaxial cable, a higher SWR might show up.
So it would be best if you tuned your antenna by using the best and shortest coax available.