Hi everyone I'm hoping for a bit of help & advice, I have gained allot of weight recently, I was never one for eating allot however I was eating the wrong things and not really exercising at all

however about 6 weeks ago I decided to make a move and try get in shape first abit about me:
Height: 5ft 6
Age: 25
Weight: 190lbs
Clothing size: 16 or 18 depending
Goal weight:135lbs
Goal clothing size (this is more important to me than weight): 12
My daily diet not is (I don't want to diet per say just change to healthy eating for good)
Breakfast: Cheerios w whole organic milk & apple
Lunch: Tomato & Basil soup & 2 piece wholemeal bread, Blueberries
Dinner: 90gm baby spinach, carrot batons, broccoli, half red pepper & 1 chicken thigh. Glass whole organic Milk
Late snack 2 handfuls red grapes & cup tea with milk
I drink 2 to 4 liters of water everyday.
Exercise wise I am limited to what I can do as I have injury to my tail bone however the last 5 weeks I have been doing 5 miles on cross trainer which was taking 35-40 mins & walking between 2-6 miles a day however I dont seem to have lost any weight on the scales , my shape has changed slightly clothes some fitting better some lose but not enough to get into smaller size as yet.
As of yesterday I have started doing 10 miles on cross trainer everyday which takes approx 1hour 10mins doing it as soon as I get up now 1 hour before breakfast(was previously doing it in afternoon) also am doing ab circle pro for 3 mins per day (I know it not what advertised however I can't do sit ups/crunches due to my back) still be walking between 2 & 6 miles per day is this enough to get losing?
I am hoping to get to 140lbs by Christmas and fit in size 12 do you think this is achievable given my plan? I'm desperate to take my son swimming he is 11 months however I currently would not be confident to wear swimming costume.
Thanks in advance xx