Though you did say that you've never bled the brakes before, bleeding the brakes is fairly elementary, and if you're not confident with it, take it to someone... and buy a workshop manual if you don't have one
If you can pump the pedal and get pressure and it stays for as long as you keep your foot on the pedal, I don't think it's the master cylinder, as the pedal would fade as you keep the pressure on it
make sure the brakes are adjusted properly, someone else gave instructions.. The adjusters are often siezed, but if yours were 'professionally' rebuilt recently then they ought to be free. The correct method is to do them up tight so the wheels don't turn, then back if off one 'notch' so they just scrape. By notch, I mean that as you do up the adjusters, you'll feel them go tight, then loose, then tight, then loose as you screw it in - this is the way it's meant to be. I'd suggest to you that jack up the front and rear, adjust all the wheels up so that you can't push them around by hand. Make sure the master cylinder is almost (but not completely) full of fluid, pump the pedal a few times and you should find that the pedal is very solid. If it's not solid, then you've got air in the system and it needs to be bled. Make sure you undo the adjusters a little after doing this
if the brakes are adjusted properly, it's likely that one of the flexible hoses going to the wheel cylinders decided to split the webbing and go all spongey inside - that can happen in one day - how you describe it sounds like it is blowing up inside until it reaches it's limit, then holding the pressure until you take your foot off, get someone to pump the pedal while you grip each line with your fingers and see if you can feel it expanding a lot, alternative is to get some vice grips and seal up the lines one at a time, check the pedal - this'll show you if one wheel has the problem
easiest way to test the master cylinder is to get a bleeder, (or even an old bit of brake line with the end crimped). take the brake line out of the top of the master cylinder, screw the bleeder in, and do it up tight, then go put your foot on the pedal - it should have a rock solid pedal, if not, then your assumption about the seals being stuffed is correct. If the seals are stuffed, you're likely to find a trickle of brake fluid under the steering column, not under the car- put your hand up the top of the pedal and search for a wet patch
you can put a seal kit through it yourself, just make sure everything is clean and goes back together the way it was. If the main bore of the cylinder is pitted or rusty, bin it and order a new one
mini brakes do need to be adjusted and bled fairly regularly, you'd better learn how to do it