There is a connection between weight and fertility but it doesn't mean your weight is automatically the issue here. As an exemple, my SIL is significantly overweight but also extremely fertile (as in, got pregnant with an IUD. Twice. Kind of fertile). So, work on weight loss by all means, but don't wait to get checked !
I understand that in the UK you have to go through your GP (sounds ridiculous to me not to be able to go directly to the gynae but apparently that's how the British system works.) so you could go to them and say, "I'm already working on my weight but I'd like to rule out other issues, including thyroid, PCOS, AMH, FSH, etc."
I also suggest your partner gets a sperm analysis. It's such a non-invasive procedure compared to many of the tests you might go through, so it's best to rule out the easy stuff first. Also I believe you have to repeat the test 2-3 times at a three months interval (that's the time it takes to generate new sperm) to be conclusive, so you might as well get it done (though tbh I dunno if it's covered or not by NHS).
Don't lose heart, it takes some couples up to 2 years to conceive ! I have two friends going through investigations atm. Struggle are a lot more common than people think, but most do end up with a child so try and be positive.
Final thing, if your diet doesn't include a lot of greens/fruit, think about taking vitamins ? I realized recently that my diet sucks (family life leaves me little time atm) and I was tired all the time. I started takings vitamins and I think I feel less foggy and exhausted. Maybe give it a shot ? I use OTC multivitamins.