Getting decent level of fitness on limited time...

If you want to get to a certain level of fitness in a limited time then you need to be specific in your training

30 mins of gym work 4 to 6 days a week
3 - 5 mile runs at the weekend

Look at what you need to improve on and work towars that
I would suggest swapping at least one perhaps two of your gym sessions into a run and actually finding some more time, once you are in the gym for 30 mins then gradually increasing that stay for 10 mins a week wont take too much out of you

As they say, if you want it bad enough then you'll find the time

I know press-ups and sit-ups don't take that much time, but they can easily be done at home whenever you've got a spare few minutes, so you could maximise the gym time available to do stuff you can't do at home instead.

Depending how many press-ups etc. you do, you might build that up in a structured way by doing something like this: http://hundredpushups.com/#sthash.6wPhfS7e.dpbs

which takes about 5 minutes 3 times a week.

The problem is, even those individuals that are reasonably fit and spend alot of time in the gym can't expect to compete with runners who have been trimming their 5K times for years. There really is a huge gap between those who think they are fit and those that can run a decent 5K time..........you'll find that out if you try and keep up with the leaders!

My pride was stung several times when I came from a background of gym work, boxing and martial arts training. The best local runners will be years and several minutes ahead of you over 5K whatever you do over the next few months. There is a huge gap between the fitness levels of those who are 'gym fit' and those who have been running and competing locally for years.

I'd say the best strategy you could use for fast results would be to simply run a fast 5K time trial every week and practice even pacing every KM. When I was starting out with running I would start races too fast and blow up. Perhaps start with a pace of 5 mins per KM and see how you do. Then drop say 20 seconds every KM each week till you find out what level you are currently at. 

I'd build up your weekend run to exceed 5 miles....but do that run at a very easy conversational pace. Doing a high effort time trial mid week is a more injury prone strategy, but honestly I've found that in the initial stages of becoming a runner young folk respond well to fast hi effort work rather than building up a base of long slow miles.......which is the usual recommended route.

I'd drop any muscle building exercises that you do in the gym and keep to hi rep toning exercises. But I'd drop a few weekly sessions in favour of a few other easy runs, perhaps with a few fast bursts. Let yourself recover the day after your time trial.

I'll probably get some stick for my recommendations. I currently follow a sensible weekly training plan, but I did thrive early on from running hard time trials at pace. I'm convinced it gave me faster results early on than a diet of slow easy paced running would have done. 

Seb Coe used to do something called 'Stage training' for the days when running was out, and gyms not available. Hotel room training.

The main exercise was squats. He'd do sets of 200 to 500 in one go.

The benefit of this exercise should not be underestimated.

Once when due to some injury, I didn't run for 6 weeks. I did this exercise and after only a few runs, went and ran 35:40 for a 10k. (The race was to close out a series)

I will start by saying this isn't a piss take.

When I was aiming for a fitness test and had to improve my upper body strength I utilised toilet time at work. Whenever I went to the toilet I added on a few minutes to the trip by doing some 'half' press ups. Ok even a dirty bastard like me ain't going to drop in the toilets and give you 25, but I would use the window ledge and do 25-50 press ups. I would also do a few tricep dips, and occasionally would do some sqauts. It is amazing if you do this every toilet trip how it starts to make a difference.

Part of the fitness tests was to do 30 press ups in 60 seconds. I managed 64 so it worked for me. Think I could manage about 10 now but that is down to using toilet time to build up strength in my wrist.

Save time spent travelling to and from the gym by running from home and doing your workout routine at home.

Run at lunchtime.

Maxpower North West wrote (see)

 Think I could manage about 10 now but that is down to using toilet time to build up strength in my wrist.

I'd have thought if you want to be competitive at 5K you'd need to run a lot more.  Instead of the gym 4-6 times a week, do a run - then a long run at the weekend.  At 25-30 miles a week you'd be a lot more competitive.