I think that if you’ve never 6x1mile before it’s better to try 5x1mile first. It’s near enough to the 10k distance to act as a guide – and with the warm-up, warm-down and jogging between efforts you still get a fair bit of miles done.
In regards to the OP, I would suggest that 3x1mile reps for a 17.30 5k runner should be in the region of 5.30 each – which is what he appears to have done.
As for joining a club and doing a Parkrun (or other accurately-measured race) both of those things are sound advice. The main benefits of joining a club are getting access to a wider range of events and socialising and picking up ideas and knowledge from other runners. But it is perfectly possible to run well and improve by just doing stuff on your own. I am currently (slowly) getting back into things after a long, long layoff. A couple of guys I know have seen me out shuffling along and keep badgering me to go back down the club but I know I am not ready for that and that if I did do that I would end up doing things that I don’t want to.
There is lots of good advice and training tips online – sometimes too much, and sifting through can be laborious but I think most people are sensible enough to spot the things that are useful, and suitable, for them.
what does the saturday session actually involve? One option is take a longer warm-up. If you normally meet up at 10.00am, or whatever, just get there earlier and go for a run beforehand. That way you'll keep up the mileage - you may be a little tired for the session but that's not necessarily a bad thing.
The hill reps are a good idea but I would be tempted to make them shorter - say 45 seconds, or even 30 seconds, the first time you do them and then increase the time if you feel comfortable.
If your easy runs end up being a few seconds per mile slower that will be fine – at the end of a run like you shouldn’t be knackered, but rather you should feel as if you could have run a little further at the same pace.
But I think you’d be generally fine to do that for three or four weeks and then adjust depending on how you’re feeling and how much progress you think you are making.