My order of priority, from your list would be:
Dishwasher - YYYY.
I wouldn't ever be without one now if I could possibly avoid it. All the people mentioning rinsing, pre-soaking or the time taken to load and unload are nuts. Scrape if there's food left on the plates, but otherwise they go straight in and it takes barely any more time than stacking them by the sink (time added well under 1 second per item). Unloading takes exactly the same time as putting them away from the dish rack after you've washed by hand (time change 0 unless you dry manually when hand washing, in which case all the drying time is saved; if you get the kids to unload that's an even bigger saving), and the whole washing bit happens by itself (time saved - massive). I do have to rewash an item very occasionally, but it's no more than 1 item per half dozen loads- and even then it usually only needs a rinse under the tap to get one fleck off, not a full wash.
Roomba - YY
For the floors you describe it's ideal. It's not great if you have deep dark carpet, lots of rugs, or lots of close-together furniture; for open plan ceramic tiles it's a marvel. You'll need to clear toys etc. off the floor before it runs round (and by 'you' I mean 'your children'). It doesn't do stairs, and doesn't get right into corners (only misses the final centimetre or so). Those bits are perfect once-a-week jobs for a small child with a handheld vacuum or a dustpan and brush. I'm baffled by the person who was put off by the emptying. Keep a bin by the charger and you just clip the back off, tip into bin and re-clip. Takes 5 seconds, and a child can empty the bin once a week. Once money is a little less tight you could also consider a scooba or braava to wash the floors as well as sweep them.
Tumble dryer (so I don't have to iron) - Maybe.
It's handy, but doesn't help that much with ironing unless you empty it and hang things up immediately it finishes. Personally I find just not ironing to be a simpler solution. Shirts go onto hangers to dry and everything else gets a good shake before pegging up.
Slow cooker - personally I wouldn't
I know lots of people who love them, but it doesn't really suit the sort of food we eat, so I don't have a lot of experience of them. My impression is that although they allow you to do other things while they are cooking, they tend to be used for recipes that need more prep time than my usual dishes, so for me it would be a time adding rather than a time saving measure. Depends very much on the sort of meals you eat though.
Other time saving things
Not a gadget, but get a big pack of cheap microfibre cloths (the ones they sell for cleaning cars are ideal). Keep one beside every sink/shower/bath and get everyone into the habit of a quick wipe round after every use. It doesn't matter how half-heartedly they swipe the cloth over, it's only cleaning one person's mess and the next wipe will get the bits they missed. If it's done regularly you'll never need to consciously clean a sink again, and the shower/bath will only need a very quick wipe a couple of times a month on the bit round the shampoo bottles that nobody bothers to move.