New consumer unit fitted - wires taped with insulating tape

No this is not correct or safe. Over time the tape will heat and become brittle causing faults in the circuit and at worse case scenario cause a fire. Sounds like a 'cowboy' has been doing this in your fuse box. The joins should be in a protective casing using specific connector block or crimps for each circuit. A decent electrician would have never done this and would have tried to not have to make a joint on the cables.

Hi, if they are just taped then no. If they are crimped and taped with a proper electrical connection then yes this is common practice. Sometimes it's hard for the electrician to do the best job for you with old poor quality cables so its good practice to cut back the damaged cables and extend them where necessary. As long as he certified the work and it looks tidy with no faults then I would not worry too much.

This is not compliant with BS7671:2008. All connections must be sound and fully enclosed in a proprietary enclosure, they can not simply be taped up and left hanging in mid air. Sounds like a cowboy has done the work. Are there certificates covering the work? (there should be a 3 page Electrical Installation Certificate, and a separate Part P building regulation certificate?) If you don't have both then its possible that you has the householder could be fined by local building control for not complying with part P of the building regulations, and fined up to £5000.