It is not actually illegal to have the Japanese knot weed in your garden. However it may be common law nuisance if it affects neighbours and has a detrimental effect of a persistent or continuing nature on the quality of life of those in the locality.
Whilst there is potentially a common law action, there is also a potential criminal action under the Antisocial Behaviour, Crime & Policing Act 2014 although it would have to be a major problem before the police or the local authority would be even remotely involved
The waste is controlled waste and must not be put in the normal green waste collection bin. It can be burnt.
You have probably read this kind of thing already.
With regard to your neighbour, I think that she has a very slim chance of bringing any claim against you .
Her claim would be in negligence. There would need to be a breach of duty of care, the duty would need to be breached and the breach and the results of the breach would have to be reasonably foreseeable. In my opinion, it is not reasonably foreseeable that her mortgage would be turned down because you may happen to have Japanese knot weed in your garden you were not aware of.
You can treat it yourself although, now they have given you notice of this, if you do try to treat it yourself and the problem becomes worse and that affects the neighbour’s property, they have a stronger claim against you.
Can I clarify anything for you?
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You don’t have to use a solicitor to reply.
My suggestion would be to acknowledge the letter and tell them that you have been advised that any loss resulting from their client being turned down by a mortgage be too remote to be recoverable.
Send them a copy of your proposed treatment plan in exactly the terms that proposed to deal with the problem including how you propose to get rid of the debris.
Tell them that your plans may change depending on the circumstances.
Mark the letter “without prejudice save as to costs” because then, they cannot produce your letter as any kind of mission you are in any way to blame.