My best experience with pest control has been the local council pest control officers in both Lambeth and Wandsworth. I had squirrels in the loft in my previous house (Lambeth) and a combination of fixing the roof (hole in soffit board) and council pest controller sorted it out.
I’ve also had mice in both properties (fairly normal in these parts, especially where there are children dropping food and all the constant building work), and council pest control good there also (and honest about what work needed to be done in terms of sealing gaps in floor etc.).
They may charge more for squirrels but for mice in Wandsworth it’s about £150 and you get three visits for that, over quite a long period.
The only thing i would say is that they use poison for mice, I’m not sure for squirrels but most likely poison as it’s the most effective for them in terms of cost and effort.