You will still need to have what ever the fault is with your line repaired regardless of which Internet provider you go with.
Unfortunately, unless you go with Virgin, all the Internet providers have to use Openreach who maintain the network and supply the engineers to carry out the repair that you need. This means you will be in the same position as now and may in fact be even worse because you will have "lost" your present slot that Openreach have provided.
Sorry aren't BT part of the same group? I work from home and need internet to do so. A 4 day time scale is not realistic for me. Thanks for the info though!
If you work from home and you depend on the Internet you should consider getting a business line which has a higher SLA (Service Level Agreement) which includes a quicker timescale to repair faults as well as a compensation scheme.
You should also be aware that logging a phone fault when your phone works OK could slow the whole process down because a telephone engineer can not resolve broadband problems. It could lead to Openreach sending out the "wrong" engineer and when he can not fix/find your telephone "fault" Openreach will either close the fault or have to send out a broadband engineer which could take more time to arrange one.