I use to have an account with them before they had the online facility, I day traded using the phone and found that and there overall service was very good.
I have not used them since as I moved to a lower spread company, that was the only downside I found with CityIndex.
It would seem the practicable thing to do when trading regularly to have more than one trading account so that you do not get left relying on the phone or no trading at all. The next issue is having more than one ISP so that again you cannot trade when it is down.
Unfortunately losing connection due to the broker/spreadbetting co or ISP will continue to happen all I would suggest is having other data available to allow you to monitor an open day trade position such as one of the free sites, bloomberg tv if trading an index and pre programme your phone to your broker for speed dialling. The alternative if faced with one of more of the above down then just don't trade and closed any open position.
These matters should all be considered when preparing your trading plan.