1. Your Mum would actually have to be convicted of an offence for it to affect her visa application to come and visit you in the UK. Even then, it would have to be a serious conviction, such as time in jail, for it to affect her ability to get a visa to visit you in the UK. Essentially, the UK Home Office don't prevent a person entering the UK where they have merely been a suspect in a criminal investigation. Accordingly, where you Mum merely got fingerprinted, there will be no adverse effects when she applies for a visa to visit you in the UK. The situation would be different if she got jail time. However, as she hasn't even been convicted of something, she has nothing to worry about.
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3. There is no link to worry about. Essentially, just because her finger prints have been taken, this does not mean that these can be used against her in a different situation as she has not been convicted of anything. The only finger prints which will feature on the Home Office database will be those of people convicted of serious crimes abroad for which there was a jail sentence.