I did this for 4 years when I lived about 3 mi (like you) from work. It was great, and I avoided tons of super nasty urban traffic in LA when I did it, but it does have some logistical challenges.

What I did-

- I showered BEFORE the run, and just used a towel to wipe down when i arrived at work. I had to walk/chill for at least 5-10mins after the run to stop sweating. I didn't need any special products/wipes; a regular small hand towel worked fine, and a face wash in the bathroom.

- I kept dress shoes at my workplace. If you can leave full clothes sets there beforehand, even better, but I didn't have that but managed to bring work attire with me in a (fairly large) backpack that worked fine for running.

- If you're running in traffic, wear proper lights and reflectives. I ran in driver-crazy LA but never got hit despite tons of distracted drivers - I used bright rear bike light, powerful front light, and 360 degree reflectives. And I never took risks at corners.

- I changed in a mens bathroom. Kinda gross, but I had no office, had to work with it, and it was totally fine.

Shower at office is great.

A good running bag is very good. I always brought my lunch and a couple other things

Leave a bunch of work clothes and shoes at work, especially if they need to be wrinkle free.

I did for a year in Sacramento after bike commuting the 2 miles for 4 years. I might start next month again in MA now that I'm not going to sit on a train for an hour

I do it occasionally as I’m 2.3 miles from my office. I’m lucky in that I have a shower and nice locker room at the office. It makes it much nicer. I’ll usually drive my car to the office on Sunday night or Monday morning with a load of clothes for the week. I then use it as a storage locker for the rest of the week.

The hardest part for me is getting my laptop to and from work. I still haven’t found a running backpack that I’m happy with. Maybe somebody will have some recommendations here.

I had good results with double run days (10k morning and 15k evening) on top of the weekly long run. I live in London as well and I found adding some miles is reasonably easy (both for cycling and running). Not sure the 5ks as the majority of runs will cut it.

Showers at work or a gym near work and a good backpack are essential.

1 A good run backpack is everything
2 Make sure you know where the washrooms are along the route. The return trip can get the digestive system working if you know what I mean
3 Drop off work clothes ahead of time. I did a 1 day on 1 day off so I could have work clothes already there
4 Start slow. Like 1 day a week, then progress to 2. I found that the backpack changed my gait and it took awhile to get used to since it changed my gait. It seemed to put more stress on my Achilles than normal running since a greater stress was put on the up and down part of the run.
5 keep up your other runs for the same reason as 4. Otherwise you get too used to a backpack and your speed will suffer
6 be visible. Your path will likely be towards work so it may not be your city's best or most common run route.

I run 8-9 miles to work, and take public transport back. (Or bike to and from, and then run for lunch.)

What doesn't work: carrying your work clothes and shoes. Leave them at work.

I don't mind not showering. I just change when I get there.

What's a good idea: know how and where to take public transport along your running route in case of injury.

I’ve ran a bunch with the Osprey momentum 32 with good success - if you do not overpack it, it is fairly light and holds the laptop very cleanly. Straps are solid and allow for chest and hip strap. Overall worked well for my 5.5mi run commute at the time.

Otherwise my tips:

1) showering is not a requirement
2) keep lightly-scented wet wipes in a ziplock bag in your backpack
3) Deodorant goes a long way