I was having a think about the site (I was there about 4 or 5 weeks ago). My guess is that the club would not want the old house and the new school buildings that are tucked up on the North and North West edge of the site.
The site is constrained on the West by the railway line (Birmingham to Stratford-upon-Avon), and the East by the Road to Stratford-upon-Avon. To the North is a private road that leads to a farmhouse and buildings (I think), and to the South is another farmhouse and fields.
The town is the North of the site.
Both existing full size grass pitches will need some work to level them and put in some decent grass. They will both likely need tall netting to stop balls going astray. There is a smaller hockey pitch, but the land it is on/in could easily be another rugby pitch. All these are to the North of the relatively new and very nice looking architecturally pleasing sports centre. I cannot see Wasps having any issues around these (planning etc).
There is, I think, an enclosed all weather pitch (hockey?) with lights and fence to the south of the centre and next to the car park (maybe up to 100 spaces). For that they would likely need planning permission to convert to a bigger 4G all weather pitch with lights, fence, etc. There is enough space between that pitch and the road, south of the car park, for another pitch. One wonders if they would build a kicking barn.
Most importantly, not much work to do, and nothing that would need an environmental survey, which someone had suggested to me just over a week ago, was a potential major issue at the Old Leamingtonians site.
They might need a change of use from education to sport. But, that might already have been done, as it ceased to be a school 3 years ago, and has since been solely used (hopelessly underused) for sport. So maybe the county would have already done that.
The interior building work would need no permissions, so would take a few months, probably done by Easter.
There might be a few external windows and doors needing changing, so a minor approval needed, but that would be a quick nod, and probably just laid before the planning committee as recommended for acceptance.
I do not think landscaping (levelling and returfing), if it does not affect any trees or hedges, needs any approval.
Ball Fences might. As would the all weather pitch. Again, chances are they would go straight through.
All in all, once the ball starts rolling, a matter of months. And a settlement of that bone of contention.
I cannot see any players who live in Leamington/Warwick/Stratford objecting. Whilst the distances are much the same, it is a much more pleasant and relaxing drive to Henley. They might find Henley a bit too sleepy for socialising though.
Those players who live in Coventry (and towns to the North) and Rugby (I think Dai lives out that way) might find it a bit more of a trek.