Existing radiators can often be retained if they are sound, correctly sized and clean, but sludge, undersizing or leaking valves should be dealt with during the upgrade.
For a East Belfast property, the answer depends on the existing heating system rather than on the boiler alone. An installer should check the current appliance, fuel type, controls, pipework, radiators, hot-water arrangement and the way the home is actually used before recommending replacement work.
For this particular question — “Will a new boiler work with older radiators in a East Belfast home” — the useful checks are the condition and age of the present boiler, whether the proposed change is like-for-like or involves relocation, the flue and condensate route, heating controls, system cleanliness and any pipework upgrades. A proper quote should spell out what is being replaced, what is being reused, commissioning, disposal of the old appliance and any certification rather than giving only a headline boiler price.
For gas work, use a Gas Safe registered engineer; for oil-fired heating, an appropriately qualified OFTEC technician is the usual route. BMK can inspect the existing setup in East Belfast and advise whether replacement, associated system work or a simpler correction is the sensible scope.