This varies heavily depending on area and price point, but overall the current market London is heavily in favour of tenants in most areas contrary to most opinions. Current rental yields in most of London are sub 1.5% for landlords pre costs, so there is generally a lot of competition to secure good, long term tenants as it's so easy and cheap to rent for people vs buying (note this is cheap purely in yield terms vs buying - London rents are still pretty eye watering for most,: its just that buying is even more eye watering).
It's also worth noting that London (and the UK in general) is a heavily saturated market of relatively unastute, baby boomer investors fearful of stocks, poor at calculating depreciation costs and who have only really seen a property bull market, who continue to think property is a great bet and continue to pile in pretty much regardless of the actual realities of the rental market, which heavily contributes to these very low yields.
As a result, Seeking out landlords fitting this profile will get you much better results than professional groups/players with 10+ properties who will be much more ruthless and yield savvy. Also looking for properties that have been on the market for a while often pays dividends - always remember landlords lose huge amounts on empty properties and will often take large reductions to fill something sitting idle - particularly if they have mortgage payments going out.
Once a reasonable place is identified, the general two key tactics when negotiating are:
stress you are a very clean, low maintenance tenant. Ask if they could recommend a good cleaner etc, discuss how bad some tenants can be in terms of long term damage/upkeep. Stress you really like the property but sadly can only afford x. You'd be surprised how much a good impression and acknowledgement and empathy with the problems they can face can make on a landlord.
negotiate long term rent freezes for subsequent years. Many London landlords are more than happy to agree to long term tenants at frozen prices initiallyfor year 2/3, and often for 3+ years in renewals if the first period goes well. Also note this effect generally rises the better and longer you have been there - do a year plus in a property, never call for maintenance and keep it clean, and you can often negotiate rent decreases in the right areas (particularly places where new to market equivalent properties are the same price you are currently paying - it is very easy to just say you prefer the look of the other local place at the same price and you have a very strong position).
It's also worth noting that doing this well is quite a lot of effort and requires a fair amount of legwork/planning, hence why a lot of people just pay the asking price and move in as soon as they need to. Most of the people who think London is a hard place to rent are just people who don't do any research, fail to plan their moving times well and in advance, or are just terrible at holding their ground in negotiations, so don't expect much back if you don't put much in.