OP, I was that weight in September, and am 5'7" and have lost a stone. I feel much better at 12st odd, though my goal weight is 11st. Sounds like you are making headway and a good start.
Though I didn't feel unwell at that weight and was quite fit anyway, and didn't have any other health conditions or issues with blood pressure, cholesterol or blood sugar, I still feel like everything is easier being a bit lighter.
Long term, sustainable changes are best and I think for now some small changes would improve things in the short to medium term. Just think of it as overall health and wellbeing improvements rather than weight loss per se.
I wouldn't worry about eating too much fat necessarily but keeping an honest food (and drink) diary for a couple of weeks - Nutracheck is really easy, good and cheap, can help you work out where you can cut down and what you might need more of. If you cut anything, cut down on sugary carbs first.
Then in the short term give yourself four nights off the wine and replace it with a cup of tea or another interesting soft drink you might enjoy.
Try and up your steps, get yourself a cheap pedometer or exercise watch. Walking half an hour a day is great - well done.
Food wise, 5 a day seems a bit of a cliché but it's a good place to start. I find sticks of cucumber, carrots and celery, cherry tomatoes and pears are quite easy to have to hand during the day. If you are cooking dried pasta , incorporate brocolli florets for the last 5-8 minutes in the same pan, depending on how soft or al dente you like it.