A few pointers on this issue:
1. The Land Registry, when first registering this plot, did so on the basis of the maps submitted to them - they did not check the boundaries on the ground or anything like that. Maps of plots for transfer were often marked onto official land registry maps. Even if they weren't, the Land Registry ultimately copied them onto their official maps.
2. For may years, the official Land Registry maps were based on relatively old O.S. data. In a pre-digital age, updating the paper maps was too big a task.
3. In recent years, the Land Registry have done trojan work in bringing their maps up to date: a Land Registry Map is now printed on the most up to date version of the Ordnance Survey. In effect, the "old" Land Registry data ( the red boundary lines on your map -which signify "registered" boundaries) has been overlaid onto the current OS data (the black lines on your map - which signify physical boundaries).
4. In most cases the "registered" boundary and the actual physical boundary are the same. But in many cases, there is some minor discrepancy. These discrepancies always existed - it is just that now, such historical discrepancies are made very obvious. That is one of the reasons why almost all Land Registry maps carry a disclaimer "not conclusive as to boundaries" ( it is possible to have the boundaries certified - though this is rarely done). That is also why solicitors would usually look for a "certificate of Site Identity" as part of their conveyancing requirements, esp. on a one-off property.
5. It is almost certainly possible to get this boundary rectified in the Land Registry relatively quickly and relatively cheaply. But as MF1 says, it is essential that you do it now. There is no point in looking for someone to blame - just accept that this sort of thing can happen, and get it fixed. Technically your house is perfectly saleable as is and you do indeed own your house ( and you do not own a chunk of the neighbouring housing estate........) - it is just that the boundaries on the LR map are incorrect; but no sensible buyer's solicitor would simply take your word for it. So you are going to have to get moving and get this fixed.
PS - I note that you are "Waiting on my solicitor to get back to me after he talks to Land registry". Some counties have had computerised land registry maps for 3-4 years now; others have just had them rolled out. Some may not yet be fully digitised. It may be that your solicitor has not come across this issue yet ( because the new digitised maps are relatively new in your county) and is more worried about it than is perhaps warranted. Don't worry about it - it is commonplace as part of the modernisation process (which is still ongoing) and these issues are ten a penny with the Land Registry at this stage. I would be particularly familiar because I have acted over the years in the sale of a lot of housing developments in rural areas, often beside 'one-off' sites, and have seen this type of thing over and over. Obviously I cannot tell you that you have no problem - but it sounds fairly ordinary and fairly capable of being readily resolved.