David,
you asked "can i hold the estate agents responsible as the house owner said that they (agents) organised an electrical inspection of the house before it was cleared for rent and passed it"
Legally, No. The agents are your landlords agents, they work for him and he is as liable to you for the mistakes they make as he would be for his own mistakes. However, the agents may have a liability to the Landlord so in short the way it works is you sue the landlord and if he wants to he sues the agent. Similarly the Landlord may have a claim against the electrician, but you don't.
Essentially people can sue in Contract (always the best from a legal point of view), but generally you can only sue someone you have a contract with. Check your tenancy agreement and I am sure you will find it is the Landlord and not the Agent. Alternatively it is possible to sue in tort (tort being an area of common law) for some things, but if you have contractual rights this is usually the way to go.
Of course you can also report the agent, electrician, etc to any trade or professional bodies that they are members of.
I should end by pointing out that it is not a legal requirement for a property to comply with IEE 16 to be let. If homes had to comply with IEE 16 before people could safely live in them then we would have millions of homeless people. Most homes do not comply, even many fairly modern homes.
In terms of your replacing the shower rather than reporting the disrepair the agents are correct in the agents are saying that you are in breach of contract, you must give the Landlord the opportunity to deal with disrepair and no court will support you doing the repair yourself if you have not done so. No matter how urgent you think it is.
I do a lot of expert witness work and frankly I think you are on a hiding to nothing. Legally the landlord isn't responsible for disrepair of the shower as it was never reported. If he had been and had not done the work quickly he would still have been doing nothing wrong, urgent repairs are those resulting in health risks such as blocked drains. So long as the property afforded another means of bathing the shower repair couldn't fall into this category. There is no argument about whether the landlord had taken reasonable steps to ensure its safety. He obviously took reasonable steps to ensure its safety by asking for an electrician to certify that it was safe, so ultimately the only argument left is was it safe enough.
Safe enough, means reasonable safe. It does not mean absolutely safe, nor does it mean complying with IEE 16. So the only argument you have left is that in your opinion it wasn't safe enough (and you might get an expert to support that view), whilst the landlords electrician thought it was safe enough (and would no doubt get an expert who supported that view. Judges hate nothing more than technical arguments between experts with "opinions", and at best you stand a 50/50 chance of winning I'd say.
Moreover you have replaced the shower (presumably you will argue it was so urgent that it could not wait). I assume it could only be urgent if you needed to use it? If have installed it as the old shower was installed you are now using a shower that you are going to try to claim you consider unsafe.....If you try that one in a court you will lose for sure. You must either think it was and is safe or was and is unsafe and if unsafe then a new unsafe shower couldn't be urgent could it?
Alternatively you may have rewired the new shower and it may all now comply with IEE 16. So you have carried out a repair that the landlord is not responsible for legally because you didn't report it and have destroyed any evidence that the installation was in any way unsafe. How do you think a court will see that?
Good luck anyway, if the landlord or agent are stupid or if you are a big enough nuisance they may give you some money. If you take it any further I reckon you'll lose personally, so long as your landlord pays for proper advice.