Do my neighbours have the right to object to the proposals in my Building Regulations application?

No. But whilst there is no actual requirement in the Building Regulations to consult your neighbours, it would be wise to do so. Regardless, you should be very careful that the work does not encroach upon their property as this could easily lead to bad feeling and possibly an action for an injunction for the removal of the work.
Objections may be brought up under other legislation, especially if your proposal is subject to approval under the Town and Country Planning legislation or the Party Wall etc. Act 1996. Building Regulations are a completely separate issue from planning permission and regardless of the need to obtain planning approval, you will still also need to meet the Building Regulations.

To see if you need approval, contact your Local Authority Building Control Department or an Approved Inspector.

No. But whilst there is no actual requirement in the Building Regulations to consult your neighbours, it would be wise to do so. Regardless, you should be very careful that the work does not encroach upon their property as this could easily lead to bad feeling and possibly an action for an injunction for the removal of the work.
Objections may be brought up under other legislation, especially if your proposal is subject to approval under the Town and Country Planning legislation or the Party Wall etc. Act 1996. Building Regulations are a completely separate issue from planning permission and regardless of the need to obtain planning approval, you will still also need to meet the Building Regulations.

To see if you need approval, contact your Local Authority Building Control Department or an Approved Inspector.