It somewhat depends on how many return lines you have and how well they are balanced as to how fast you get hot water, but, yes, done right, it is almost instantaneous. Some tankless systems should not be run with a recirculation system, but if you are allowed, then there are specific rules on how it is to be done. You may need to add a small buffer tank to do it right and not have the heater running all the time there's water circulating.
Replacing pumps may be easy and not require any soldering at all, depends on how it was installed. On some, there are gasketed connections, and you just unbolt the thing. Assuming you get the same pump, a new one should bolt in (maybe with new gaskets). The electrical connection normally requires a screwdriver and maybe some wire nuts.