Get rid of screens! Or at least limit to maybe one family film a week. My family didn't have a TV but we were lucky enough to have a big playroom and garden so us kids did lots of active play instead and were very fit and healthy weights, despite eating massive adult portions at every meal. My mum couldn't drive either so we walked to school, about 20 minutes each way, and to town, about 1/2 an hour. We walked or cycled to different parks every day in school holidays and swam every weekend.
Don't let the weather put you off! Just get waterproof overalls (for you too!) and wellies, and I don't know if they still sell them but a couple of years back JoJo Maman did these great waterproof hats for little ones, with a soft fleecy lining. Hit the playground for half an hour every day after school. One of the perks of breastfeeding is the milk is always right there if your tiny one gets hungry out and about! Sterile, just the right temp, all ready to go!
At the weekends and in the lighter months go for walks together. If he's not keen here are some ideas to liven them up:
Each think of 3-5 things you're looking for (maybe write them down and he can cross them off when they're spotted?) like 'a brown dog', 'a man on a bike', 'a pink flower' etc etc. Or teach him how to make a tally chart and he can count different coloured cars, or how many trees, or dogs, or whatever. Tell stories as you walk, or play games like 'I'm thinking of an animal' or 'Would you rather...?' Or play imaginary games like you're arctic/jungle/ocean explorers as you go. Or write down the letters of his name in block capitals and he has to cross them off one by one as he spots them in signs or street names or car numberplates (only one letter for each sign though!).
Stop buying crap, if that's a thing you often do. No more fried food and ready meals. You can get your little boy to help prepare a couple of vats of healthy something-or-other-that-freezes-well at a weekend and freeze a portion or 2 of each of those for later in the month. For quick meals on weekdays you can still have healthy stuff that just doesn't need much input - baked sweet potatoes and broccoli, English muffins with peanut butter and banana slices, scrambled egg with peas, houmous with veg sticks and baked pitta strips, spinach tortellini and pesto, stir frys... that sort of thing. Literally ready in minutes but healthy. Or get a slow cooker and get it going in the am. And if he's used to puddings gradually swap them out for Greek yoghurt and fruit, maybe a chocolate covered rice cake after.
When the weather is really gross and you are stuck indoors, you can still do fun active stuff together. Maybe the Joe Wicks workouts? I lost 3 stone with him in the first lockdown, just doing the simple kids ones! Or you can have treasure hunts around the house (Not for chocolate, but for his favourite teddy or, if you want to be educational, shapes/numbers/letters) or set up a simple obstacle course where he has to jump from cushion to cushion and not touch the floor. Or have a living room disco with the lights turned down but flashing fairy lights draped around, and let yourself go to some cheesey 80's music! Depending on your budget/space you could get a piklar triangle or indoor climbing frame/slide that folds away, and comes out at times like these? I have a folding slide and inflatable ballpit which is just fantastic, especially when they have friends round. It can all be stored away in the garage the rest of the time. If you have a nice straight not-too-steep staircase you can turn it into a fab slide with thick sheets of cardboard (ask at Currys if they have any old fridge-freezer boxes you can take) and a duvet at the bottom to land on. Running up the stairs for the next go is great exercise! But I think tackling the food is the biggest thing. I think my best tip is NEVER SHOP WHEN HUNGRY! If I do that I end up buying a tonne of sugary crap that I'll hate myself for later, but if I go after eating I stick to lovely sensible healthy options! Maybe to make things fun you could each choose one new fruit or vegetable that you didn't have last week and look up recipes online that include them, that you can make together. Buy or make a chefs hat and apron for him and turn it all into a big game. Maybe it could be something special the 2 of you do together while someone else has the baby, a bit of special one-on-one time?