Something sounds odd to me here. There is only a one wire signal from the heating that tells the boiler to operate. Then the boiler circuitry starts the fan, ignites the pilot light, checks that its lit and fires up the burner. Hive doesn't do any of that, it's all down to the boiler.
When you installed the Hive, the boiler wiring shouldn't have been disturbed at all, so nothing should have changed there. Was it working correctly before the Hive was installed?
If the Hive is starting the boiler fan, but the boiler isn't igniting, that suggests that there is a problem with the boiler, but if that was the case, it wouldn't be working properly when the hot water switches it on, which you say it is. So, are you absolutely sure that the Hive causes only the boiler fan to come on?
Here are some things I would consider.
1. What normally goes wrong is with the removal of the original room thermostat. It cannot be just disconnected otherwise the heating wiring will remain 'open circuit' and the heating won't come on. So, the old thermostat live switching wires need to be bridged to complete the circuit. However, if this is the case and your description is of what is happening is correct, nothing would be coming on, including the boiler fan.
2. You don't give any details of the type of heating system you have. Do you have an very old system with pumped central heating and gravity circulation of hot water? If you do, the Hive needs to be set to being Gravity / Part Pumped.
Don't use this setting if you have a fully pumped system, or one that has motorised valves.
3. Have you waited a minute or two for the boiler to ignite in case it is still hot from heating the hot water?
4. Have you done any other work at the same time? An airlock in the radiator pipes could stop hot water leaving the boiler and so the boiler wouldn't ignite.
I see it's your first post so welcome to the forum. For future reference your post would have been better in the Plumbing and Central Heating forum were the experts in this field hang out.
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"This forum is for posts about disasters you have experienced. General questions or queries should be posted in the appropriate forum section"