I am interested in which law says that fixed term contracts are only legal in specific circumstances. The reason being that in 40+ years of representing members, I have never come across that law - despite fixed term contract being common in my area of work. Could you please reference the relevant law?
OP - you have an email, so you don't need a new contract. The other terms remain as stated originally, it is just the end date that is varied.
You are working in a local authority, many of whom have (a) budgetary pressures and/or (b) "funny money" - external funding that is time limited. Therefore fixed term contracts like this are commonplace either because the budget for certain areas of work cannot be guaranteed, or because the money itself is time limited and they have no control over that. At a guess, I would say that your post is very likely paid for under (b) - in these circumstances posts are often extended for a variety of reasons which may include underspend on a budget being allowed to carry over, additional funding becoming available from the source or (linked to the latter) a change of policy or government means that the government /funder wants to "change or review" their spending and they allow an interim continuation whilst they get their new ducks in a row.
Fixed term contracts are virtually the norm in much of the public sector now, but if this is your only reason for concern, then I would advise you just get on. In local government they have pretty much as much security as any other post - they can be extended, or they can be terminated with proper notice. "Permanent contracts" are only as permanent as the length of your notice period! If you have seen the redundancies etc in the public sector now you would understand that there really is very little difference in terms of security.
In almost everything the law treats people with fixed term contracts exactly the same as other employees. The same is true of most local authorities, and that is often better than the law does! When contracts are coming to an end, whether or not you have two years employment, most authorities treat you as redundant and offer exactly the same redeployment opportunities as any other employee would get. Once you hit the two year mark you are also entitled to redundancy pay and whilst that is true for all employees, local authority redundancy pay (if not redeployed) is often paid at a higher rate than statutory.
If you like the job and/or want to remain with a local authority, chill and stop worrying! You can apply for internal positions if something you fancy comes along. If it doesn't then all things being equal you have a years work, and a good chance of potential redeployment rather than unemployment. I think you are overthinking this - probably because you are relatively new to local authority work? What you are describing has been the norm for well over a decade, in some areas of work 2-3 decades! It is significantly less worrying that people often think. Local authority culture is significantly different from the private sector. It isn't perfect, but they tend to be honest. If they have only two years funding for a post right now they use fixed term contracts and tell you. More often than not the private sector don't because it's fine - they can get rid of you whenever they want anyway, so why bother with a little thing called transparency?