You’ve taken the right step by bleeding the radiators first; it’s almost always air in the system when radiators aren’t heating up properly. There are two other possible reasons for your radiators not to heat up, balancing or a pump failure, and the second is the most likely.
Balancing Radiators
Balancing is a tricky art and really needs an experienced heating engineer. Because hot water cools as it is sent round the radiators, the lock shield valves (those are the smaller of the two valves at the end of a radiator) need to be set so that it is harder for radiators closer to the boiler to get water and easier for those at the far end of the piping run. Done correctly, this will mean that the radiators furthest away from the boiler should get as hot as ones closets to it.
This is less likely to be the problem with your system though. If the boiler is upstairs, then it could be, but you would almost certainly get patchy warmth in some of the downstairs radiators.
Diagnosing a Pump Failure
If there is a really big contrast between the upstairs and downstairs radiators, like all the upstairs ones are hot and the downstairs ones are stone cold, then a damaged pump is more likely.
You should be able to find the pump close to the boiler, or the hot water cylinder if you have one. It will be a lump about 3 or four inches in diameter attached to one of the pipes. It will have electric wires running to it and they are often red in colour. Put the central heating on, turn the thermostat up to maximum and give it five minutes. Then feel the pipes either side of the pump (carefully!) and if one side is hot and the other cold, the pump is faulty.
It is possible to replace a pump yourself but unless there are isolation taps wither side of it the whole central heating system will have to be drained down. Also sludge and air in the central heating system can potentially damage the new pump. Bearing these factors in mind it may well be the most sensible option to call in a qualified heating engineer.
