I wouldn't want a washer/dryer in the living room if I could help it. What are the dimensions of what looks like a linen closet in the bathroom? That location would be close to existing plumbing. Some compact stackable washer/dryers are only 27 inches deep. If you took the door(s) off of that closet, would you have 27 inches of depth? If so, maybe the too-small space at the end of the hallway could become a replacement linen closet (built in place or a freestanding storage unit)?

Start with a floor plan, 2 dimensions, black on white paper, no color, every wall, window and passage has a dimension in inches,. No to the living room for a washer and dryer. I've no clue how two "bedrooms" are crammed in the corner.

+1 to posting a floor plan.

Is there room to add the W/D in what I am guessing is the front entryway/breezeway to a garage?

Do you have a finished basement or heated garage that the W/D can be located in?

Are you willing to combine the two small bedrooms to carve out more hallway space for the W/D?

My first choice would be the linen closet in the bathroom if you have the depth. I would look at a stacking apartment size which take up less space. Plumbing is there and hopefully you can get electric there. Also if it is deep enough and you can access it instead from the hallway again you can see if you can open it up from the hallway into that same linen closet. Also look into redoing that add on room with insulation, electric and plumbing and put it in there as a mudroom/laundry room set up. That would be my preferred space.

What about a small additon off the kitchen that could be both laundry room and pantry?

First choose an all in one W/D and some now do not even need venting so opens lots of choices but in the kitchen would be obvious these are issues in most homes in Britain and why so many have the laundry in the kitchen

I’d rework the white extension for laundry since it sounds like it needs to be anyway

Where are the water heater and HVAC system? If they are out in the unpermitted extension you intend to tear down, putting them and the washer/dryer in the replacement addition would make sense.

That bathroom cupboard is 18" deep, could be bumped out to 21" deep max.

How many people will you be doing laundry for? The Europeans have some very small models, but, needless to say, they don't hold a lot, either. Hubs had a company apartment in a European city with a very minuscule washer in the tiny bathroom. Running electricity to your extension would probably be cheap and if you need bigger appliances, you could put your dryer there.

I would probably put the washer in your kitchen on the wall with the tub (cheap to plumb) and move some of your pots and pans or whatever else you have in the cupboards to the extension. Or to your living room cabinet. Usually, people have things they use daily and things they use less often. Think about what things you could exile from your kitchen with the least inconvenience.

How large is the window in the kitchen? Could it be built into a cabinet either on the window wall or next to the entry door that comes in from the extension?

Is the sink next to the dishwasher? Could you move the sink and dishwasher to the wall with the window and put the washer and dryer on the dishwaher wall.

You should be able to get to the plumbing since the bathroom will share a wall. You will need water and sewer for the washing machine anyway.

Maybe make a L shape design with the sink, dishwasher, and washer/dryer.

I would place it in the kitchen or bathroom. There are new models out there that are “all in one”, ie just one unit. Not too sure how they are but might be worth a look. OR if there is a basement put them there.