The biggest obstacle to visiting a lot of places by bike, especially while touring, is having nowhere safe to put your bags. The Eden Project is the only place I know that has very prominent and numerous lockers for exactly this purpose.
The WWT centre at Barnes had (and I think still does have) some good cage type bike parking where you can leave your bike (up to 2 per cage I seem to remember ) plus any bags and secure the cage with your D lock. That seems like a good way to handle the sort of luggage you might have on a touring holiday.
We have had some very helpful experiences at National Trust properties when asking to leave bikes and bags - usually at least they have been happy to keep bags behind reception, but great service at Moseley Old Hall where bikes with panniers still attached where whisked off to a locked shed by a very helpful volunteer.
Our worst experience was a few years back at Alnwick gardens, not actually when cycling but on a walking/bus travel holiday (due to a broken wrist) - which presents similar challenges for luggage as we didn't want to lug large rucksacks around the gardens. Early negotiations to leave the bags were scuppered by a rather zealous supervisor who told us it was not possible (probably health and safety or maybe security concerns) so we got our admission refunded and left without visiting. A letter of complaint though was answered promptly and helpfully with an offer of a free visit in future and an assurance that they understood the issue and would put in place a solution. Unfortunately of course we did not visit the area again so couldn't check whether they had sorted it.