This is an interesting one
It may seem an odd place to start but you do not have any kind of contract with either the farmer or the council. As such you will be relying on whether they owe you a "duty of care" in tort.
There are several situations where a duty of care is firmly established such as between road users.
Starting with the farmer you will find it difficult to establish he owes you any duty. There is a case called Rylands v. Fletcher where a farmer failed to maintain his pond which burst flooding the land of his neighbour . He was successfully sued. However there are differences here. First should the farmer plough across and not down the water will not drain and is likely to ruin his cross. Secondly, the rain is a natural occurrence and passes over the land owned by the council to get to you.
The council have a duty to provide drainage. The drainage must be adequate for normal rainfall. As such if there are monsoon type conditions and flooding occurs the council are not ordinarily liable. However , if they are aware of a particular problem, a flooding "hotspot", they take no action and flooding occurs they are potentially liable. As such they should do something to keep the gullies clear. This may be to put more in or check existing ones more regularly.
Finally I suspect that the road was there before the barn was converted. The builder will owe you a duty of care and the HSBC agreement may still be in place. I would get a surveyor to look at the drainage on your property to see if it was adequate given the position of the barn and a quote for beefing it up.
I would wait to see what the council propose but getting the drainage beefed up may be the cheapest and quickest solution. It would also mean that you are not dependent on someone else doing something , like clearing gullies more, for your home to be safe.
I hope this helps, please reply if there are any further points and I will be happy to respond