Forgot to add that my Dcs are almost exactly the same age as yours, OP - so I can relate with your dilemmas.
I loved being an expat, have been one for most of my adult life, and in different countries, but still would advise against expatriation in France.
About language, 'learning French' or 'speaking French' is different from mastering a language, being able to crack some jokes and understanding the locals' sense of humour and what underlies it, share a proper conversation beyond asking for a baguette with a charming accent that everyone will find melting (like every single French people, I absolutely adore the British accent) - of course people will be friendly if you try to speak a few words. Even friendly tout court in most cases. But it does not mean in the least you will integrate. Mastering a language, which means a deep familiarity with the culture that produced it, takes many, many years, it is hard work, and a long term personal investment.
I have to admit I am deeply biased in favor of the UK though - lived in many different places, included villages, but NEVER had issues meeting people and making friends. I cannot stress this point enough. I have always found that people were quite nice and trusting (whether you are a church goer or not, amazingly :) .
I experience the exact reverse here: people tend to be suspicious first, it is incredibly difficult to simply chat. And that, in itself, can be isolating very quickly.
One example - and I think it is quite relevant for the OP: when I lived in the UK, I attended a few playgrounds. Every single time I did, I ended up talking to other mums, effortlessly. We just chatted, starting with our babies, then a bit about ourselves, ect. Well, chatting, you know. Having a bit of a social life, even if just for a few minutes.
Ever since I arrived in France, and that is more than 8 months ago, I did go to many 'ludothèques' (which are usually very nice, huge rooms where you find lots of toys for children of various ages), sat there while my DCs were doing their stuff, alongside other mums, and no one said a single word to each other. It invariably struck me as being absurd: sitting there, with other adults, watching our kids, and never uttering a single word to each other. I tried once or twice to initiate a conversation but it died off very quickly - anything beyond: 'what age is yours' seems to be frowned upon.
Unsurprisingly enough, there is only one occasion when I managed to properly 'chat' with another mum: she was French, like myself a ex- expat, coming back from the US, where she had lived for 20 years, and finding it just very hard to adjust to France.
Being a mum to two young Dcs can, in itself, be a bit of an isolated experience if you don't work. Being abroad can multiply this isolation exponentially, and it is bound to affect your experience so it is also something to think about. I would carefully look at your job options in France before making the jump for that reason, too.