Building on what bailey bird correctly says, there are 2 issues and they are separate.
Convenents are placed on a plot of land by the seller of it - and generally only a seller can remove it. If the seller (who as the owner of another piece of land) placed the convenant in such a way that his successors in title (people buying his land after him) control the convenent, you may well find that a present-day landowner somewhere locally has control over the convenent. More likely you will find that there is no trace, or the convenant was in favour of a person long since dead. In such a situation, a new landowner could apply to have the convenant removed.
The second and separate issue is planning consent. Whether or not planning consent can be granted depends on local planning policies. If you live in a built-up area (one with a settlement boundary around it), planning consent is normally grantable, subject to other polices (for example, similar size and scale of housing to that already in place. However as next-door neighbours you would be consultees in the process and there are things you could do:
1) Teramb's idea (or similar)
2) Call the Tree officer and enquire about getting a TPO on the big oaks - at least that would protect some of what you enjoy as well as constraining what might be buildable on the site.
3) Ensure you write and make it a planning condition (if granted) that the underground spring is properly dealt with to protect your land.
All of this supposes that planning has been /is being sought.
I suspect that this is a 'difficult site' and planning is going to be difficult to get. I say this because most site owners will get OPP BEFORE they market the land- its worth a heck of a lot more money when done what way.