United Kingdom

Yes.

Applicable law:

• Regulation (EU) No 910/2014 on electronic identification and trust services for electronic transactions in the internal market' (eIDAS).1
• The UK implementing regulations: Electronic Identification and Trust Services for Electronic Transactions Regulations 2016 (SI No. 696). These regulations updated and amended the Electronic Communications Act 2000.
• Electronic Communications Act 2000, Part II. Section 7 states that an electronic signature is admissible in evidence.
• See generally the Law Commission, Electronic Execution of Documents, No. 386, September 2019, which states: "An electronic signature is capable in law of being used to execute a document (including a deed) provided that (i) the person signing the document intends to authenticate the document and (ii) any formalities relating to execution of that document are satisfied."

Types of electronic signatures:

The eIDAS recognizes three types of electronic signature: (i) simple e-signature (SES), (ii) advanced e-signature (AES) and (iii) qualified e-signature (QES). SES can be any form of electronic message associated with a natural person (this can include typed signatures, e-mail blocks, etc.)  AES is an electronic signature uniquely associated with an individual and linked to data, so that any subsequent change in the data is readily identifiable. QES is generated by a qualified electronic signature creation device (backed by a certificate issued by a qualified trust service provider) and has the same validity as a handwritten signature.

While QES has the status of handwritten signatures, there are no documents that require use of a Qualified Electronic Signatures in the UK.

Notable case law:

In J Pereira Fernandes SA v Mehta [2006] EWHC 813 (Ch), it was determined that the automatic inclusion of the sender's email address after the document had been transmitted does not amount to a signature under s. 4 of the Statute of Frauds.  In this case, the court reasoned that an automated process generated by an application (e.g. the sender's email address) did not evidence the necessary intention to authenticate the document on behalf of the user.
 
In Kathryn Bassano v Alfred Toft [2014] EWHC 37 (QB), it was determined that a regulated agreement under the Consumer Credit Act 1974 could be signed electronically. Here, the court simply confirmed the liberal approach to what is considered a signature under UK law.

Yes.

It is generally considered that wills cannot be executed electronically.  Documents requiring witness attestation or public filing (e.g., deeds, land registry documents (including leases over 3 years), revenue and customs documents, family law documents, and corporate accounts (where stamping is required)) may require the physical presence of witnesses or other formalities, even where the signatory and witness may be using an electronic signature.

Yes.

The Electronic Identification and Trust Services for Electronic Transactions Regulations 2016 (SI No. 696). The Office of the Information Commissioner is the appointed supervisory authority under eIDAS. You can read the ICO's eIDAS guide here. 

The eIDAS Regulation, which is directly applicable in the UK, provides for the general legal framework for qualified trust services. Among other things, the eIDAS Regulation governs the application procedure for trust service providers to obtain the status of a qualified trust service provider (Art. 22) and the requirements applicable to the same (Art. 24). 

Yes.

Accreditation for trust service providers in the UK is provided by tScheme, a self-regulatory body, which maintains the UK's list of trusted services here.