What is actually happening here is that the client is attempting to (re)define the contract after you have fulfilled your part of it. This is common, either out of ignorance, presumption, or outright bullying and is generically known as a 'copyright grab'.

Legally, contracts and variations of contracts cannot be imposed, they are an agreement between both parties (unless one party already signed away the right to object). Since you made a verbal contract to do the work, then did it to the client's satisfaction, it is too late for them to now require terms they failed to specify. So, no, you are not obliged to accept their proposal.

However unless you have documented proof of what was agreed, eg signed T&C's or at least emails or letters or notes or recordings of phonecalls, what actually was agreed may be disputed by the client. Clients often mistakenly believe that they are the owners of copyright in commissioned work, although frequently it is a deliberate attempt to  twist your arm to obtain more than has been paid for. But, in the absence of an agreement you own copyright as the creator of the work. Presumably you based your pricing on this understanding, too. The onus is on the client to prove otherwise, not you.

The diplomatic way out of this may be to point out that a copyright buyout will be very much more expensive than your quoted price for first use. Almost certainly they don't need the broad rights they are now asking for and will not want to pay this. If they do, they should pay accordingly, and the Association of Photographers recommend that a buyout should cost somewhere around 7.5x-10x the first use fee. You may be able to agree a license that includes the rights they really need at a price that is mutually acceptable. If not, then stick to the first rights fee for first rights.

Don't be intimidated by threats such as 'you'll never work for us/this industry/in the UK ever again'. Clients who don't play fair are not worth keeping.