So my view is that due to MTV's broadcast, this has clearly damaged your reputation. It is difficult to know what the damage may equate to in monetary terms - defamation cases are difficult at the best of times but generally, if the press or a broadcaster makes untrue statements then they would be found liable for your loss, now and in future. So you could have a claim for stress, loss of earnings, loss of future earnings too.
A defamation claim requires that someone has either verbally said something (slander), or written something (libel) which was untrue and they knew it to be untrue - with the intention to damage your reputation. The untrue comments have to be brought to a third party’s attention to qualify.
There is a defence if the person who made the comments can prove the comments were true, or they held an honest belief their comments were true.
So there is a high evidential burden to prove defamation - meaning a lengthy court case. Unless they admit liability early on which is desirable - and the law firm would push for an early admission so that you could secure interim funds before the case finished.
You can claim for any financial loss incurred such as lost business, lots future job prospects and anything else reasonably incurred.
The limitation period is only 1 year for a defamation claim, so the law firm will need to ensure they issue the claim at court within a year of the incident.
To stop further damage either yourself or the law firm would need to send them a cease and desist letter. A template “cease and desist” letter is available at https://www.lawdepot.co.uk/
I would ask the law firm to this - whilst you could send it to MTV they have a legal team and you should instruct one of your own so you are on an equal footing.
These claims are complex with them being heard in the High Court so you would be well advised to instruct a firm on a no win no fee basis - a few do offer this so you would not have to pay anything up front. Here are examples of law firms who could help you with this, though there are others :