I'd go- make sure you take duvets aswell as sleeping bags and hot water bottles in case it gets cold. Head south- it will be warmer- New Forest/Dorset area (near enough to rush home if it gets too rainy/cold)
If I was you, I would stay really close to home and go to Wellington Country Park. Nice friendly, well-organised site with clean facilities, nice walks, massive sand pit for DD, lovely toddler's play area, and the opportunity to come home easily if it pours down.
(but then I recommend this place all the time because it is my favourite and my best).
though doesn't have showers.... another place that is recommended not too far from you is
took ds on his 1st trip (16months) mid sep couple years ago and he slept in a travel cot which was on a picnic rug wearing 2x babygrows a grobag with arms (think from jo-jo maman?? as could use for travelling or bits of fleece really badly stitched also do the trick)in his cosy toes from the pushchair wearing a hat and mittens...with a couple of fleece blankets to hand if needed...he was fine although did wake one night as massive storm and just came in with me ...which is what still happens now he's 4
the first night is still hard as out of the norm but the fresh air usually does the trick after that ... have learnt not too fret about sleep/anything as does not impact when we get back go straight back into routine..
would highly recommend getting good waterproof dungarees and jacket for playing in the rain/mud whilst keeping warm
Have camped 3rd w/e in September for at least last 10 years (last 6 with DC) DD was 7mths when she first went and DS 10mths. It has never been too cold and last year was the only good week of the summer
Found all-in-one fluffy suits great for keeping DC warm in bed and more recently Primark fleecy pyjamas though grab them whilst you can in autumn for the following year!
Definitely go and have a wonderful time - my two (now 6 and 3) LOVE camping and already have so many great memories