I was able to turn my lawn without "replacing" it with the following fairly simple process. Note that when I bought my house, the previous owners had all but given up on yardwork. I don't think had done any all year, and I signed papers on July 2nd (of many years ago).
So first is mow it and then keep it mowed. For the super tall stuff, you may want to use a weed wacker to cut it down.
Every year, I put down a fertilizer with weed preventer and crab grass pre-emergent. I put this down once in the early spring then again mid-summer. Crabgrass is a big problem here in New England, not sure about Old England
Then I try to overseed areas that need it in the middle of winter. My theory is that the seed will stay dormant. Then when the snow melts, it will get nice and germinated from that moister and start growing right away and be established enough before i put down my weed preventer.
I did that for two or three years and it really turned the whole yard around. It helped that one of the years was extremely rainy. Seems like lately, most summers in Massachusetts are droughts. But your yard looks small enough that it should be easy enough to water.
You can accelerate the process by digging up bad spots and pulling weeds by hand and planting new seed. But if you just do the basics and are in no rush, you'll be tipping the tables in favor of grass and over time, your lawn will start to really thrive.