I've asked older people, family members etc. if they thought the climate was changing. Everyone has said yes, they think the seasons now "begin later" and the weather is more erratic than it was in their youth.
I've noticed certain invasive species increase in number in the UK over my lifetime. When I was a kid I didn't see Fallopia japonica all over the place, now it's very common and very hard to get rid of. DEFRA seem to think introducing a Japanese beetle to these islands that eats it will get rid of it. The same can be said of Dysdera crocata, that I first saw when I was 15 years old and now see everywhere, including a new melanistic variety that seems more common than the original. That said, I don't think this is down to climate but to a lack of natural predators and a habit of propagating both vegetatively and sexually in the case of Knotweed, and the natural highly aggressive behavior of Dysdera crocata gives it an advantage over native spiders and prey. Catch one in a bottle and the hyperactivity and aggression of these spiders is evident when compared to any native British spider.
I see a lot more crows around, again I put this down to the ability of crows to adapt well to any temperate environment above anything else. I have seen a decline in seagulls. I remember as a child visiting the area I live in now and seeing more seagulls. Seagulls do feed on fish which have been depleted in some UK waters but again the seagulls will also feed on landfill and tourist handouts, and they have not been supplanted by any other bird species in the area so the lack of them isn't easily explained. A disease perhaps.
I have noticed far less mosquitoes than when I was a child, despite living near a body of water now, which while personally welcome may be impacting the local ecology. I don't think I've suffered a single mosquito bite in years despite some hot humid summers.
I grew up in the North West and this is the South West with a distinctive ecology and geography set apart. The area I'm in at the moment does appear to be very healthy (it's a protected coastline) and I've heard certain rare birds and insects have returned to the area. Since I dive quite a bit I have noticed some interesting blooms of jellyfish lately that is out of the ordinary (such as several thousand Man O Wars ending up on our coast due to storms, this is not normal) but I've not been living here long enough for any subtle changes in marine flora/fauna to be obvious to me. Judging by the roadkill numbers there seem to still be plenty of fox, badger, rabbit, pheasant hedgehog, deer etc. being supported by the land.
Living on the coast, the weather is consistently changeable; it's when it gets inland that it appears to people to have changed - longer periods of rain causing flooding or longer spells of sun causing drought, which doesn't affect my area as much as others. You can generally tell when something is abnormal, since the UK councils are always totally unprepared for it, be it a drought, gale or a snowfall.
If I had a garden, I'd no doubt be able to see some changes there if any, but I don't - all I can do is observe what I see when out walking the area. The Cornwall area is known for moors, gorse and heather, so it's a fairly specific habitat with it's own population of adders, slow worms, lizards, etc. which don't occur elsewhere in the country. I've noticed some insects I've never seen before here like the Hummingbird Hawk moth, which don't normally hang out in the UK but prefer the Med. Could be a sign things are warming up here. The area's seen an increase in birds of prey which certainly means their prey - usually smaller birds - are increasing.