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I'm in the same situation, moved 2 of our houshold Mobiles from Vodafone to BT mobile to only find that we have no coverage from inside the house. The coverage map shows that we should have and we have never had an issue with the Vodafone signal.
I've heard that your BT mobile will work with the EE signal booster however the issue is getting the EE signal booster box activated by EE so it can be used, as they currently will only activate for customers using EE, Orange or T-Mobile, not sure how much truth there is in this and was wondering if anyone had any first hand experience?
Just for information, I now know that you need a Pay Monthly (contract) with either EE, Orange or T-Mobile in order to activate the Signal Box, a Pay As You go will not work. The signal box needs to be either new or if second hand, de-registered, this is not EE being over zealus, it's because they are strictly controlled and regulated by Ofcom in the UK.
Fortunately my step-daughter has an EE contract and as she splits her time between living with us and another address and as such she was able to activate the box.
I can confirm that once activated that it works like a dream with BT Mobile.