This is a good place to start with what you are allowed to do etc.:
https://www.planningportal.co.uk/
more specifically here:
https://www.planningportal.co.uk/info/200130/common_projects/43/outbuildings
Outbuildings are considered to be permitted development, not needing planning permission, subject to the following limits and conditions:
- No outbuilding on land forward of a wall forming the principal elevation.
- Outbuildings and garages to be single storey with maximum eaves height of 2.5 metres and maximum overall height of four metres with a dual pitched roof or three metres for any other roof.
- Maximum height of 2.5 metres in the case of a building, enclosure or container within two metres of a boundary of the curtilage of the dwellinghouse.
- No verandas, balconies or raised platforms (a platform must not exceed 0.3 metres in height)
- No more than half the area of land around the "original house"* would be covered by additions or other buildings.
- In National Parks, the Broads, Areas of Outstanding Natural Beauty and World Heritage Sites the maximum area to be covered by buildings, enclosures, containers and pools more than 20 metres from the house to be limited to 10 square metres.
- On designated land* buildings, enclosures, containers and pools at the side of properties will require planning permission.
- Within the curtilage of listed buildings any outbuilding will require planning permission.
So - your restrictions are single-storey / max eaves and overall height (2.5 and 4m - if more than 2m from a boundary) / no more than 50% of the plot other than the house
Permissive development encompasses a wide range of options, and you can in fact build quite large buildings without requiring planning permission - as mentioned above, there are restrictions on what you can do, e.g. roof height for a pitched roof is a maximum of 4m in height - and if you had one pitched roof, that might not work on that size building - however there are other options, so you could have a multi-peaked roof - hard to describe, but imagine building a series of sheds next to each other your roof would go up and down - now take out the walls and you would have one large building with the correct roof pitch, and height etc - a more complex roof and you would have to deal with drainage in the gullies - but it is possible...
based on that...
you want 30ft wide in an overall 40ft width - this would leave you 5ft from either boundary - that will restrict your roof height to 2.5m if you knocked off an extra ft+ each side to bring it to 2m from the boundary, then your roof height can be 4m - it will make a big difference. Other than that - you are just under 50% of the back garden, so if you are single storey and can keep the roof height as required, it would suggest that you can build this on the basis of permissive development and you will not need planning... I think that as a neighbour I would still object (may have no grounds, but...
) alternatively I would be over playing with your tools and joining in!
you will still need building regs - but that is HOW you build it not CAN YOU build it...