I don’t think you’ll have to isolate while your family quarantines although it’s remarkably hard to find a proper answer to this on the government website. As you’ll be double vaccinated you’d presumably be allowed out even if they tested positive (provided you had a negative PCR and were symptom free) so seems unlikely that you’d need to isolate while they undergo their quarantine period
I don’t think you’ll have to isolate while your family quarantines although it’s remarkably hard to find a proper answer to this on the government website. As you’ll be double vaccinated you’d presumably be allowed out even if they tested positive (provided you had a negative PCR and were symptom free) so seems unlikely that you’d need to isolate while they undergo their quarantine period