Yes, they can litigate and arbitrate at the same time. It may not be a classically Good Idea but the commercial disputes cited below show that it happens.
The court does have an inherent power to stay proceedings. In the Court of Appeal, Ahmad Al-Naimi v Islamic Press Agency [2000] EWCA Civ 17, Lord Justice Waller said (my note in square brackets):
A stay under the inherent jurisdiction may in fact be sensible in a situation where the court cannot be sure of those matters [= the existence and scope of an arbitration agreement] but can see that good sense and litigation management makes it desirable for an arbitrator to consider the whole matter first. If, for example, the court thinks that it would take a trial with oral evidence to decide whether the subject matters of the action were actually within the scope of an arbitration clause, but that it was likely that on detailed inquiry the subject matter of the action will be found to be covered by the arbitration clause; and particularly if an arbitration was bound to take place in relation to some issues between the parties, and where having explored the details necessary to found jurisdiction, it would only be a short step to deciding the real issues, it will often be sensible for the court not to try and resolve that question itself but leave it to the arbitrator.
The inherent power is in addition to the possibility of a mandatory stay under section 9 of the Arbitration Act 1996, which implements Article II.3 of the New York Convention 1958. Section 9 provides, in essence, that one of the parties can apply for a stay on the basis that the dispute is covered by an arbitration agreement; but it doesn't require the court to do anything proactively. The existence of section 9 and the general tenor of the Act have led courts to treat the inherent power as something to be deployed in exceptional circumstances (see Lombard North Central below). Attempting to litigate the exact issue that's simultaneously being arbitrated could be such a circumstance, but there are also patterns that are less neat, and where the parallel processes can continue.
There has been much litigation about exactly how connected to the arbitration agreement something has to be, in order to justify a section 9 stay. Clearly there are times where two parties are engaged in arbitration about some matter, but separately appear in court over a completely different topic. When the parallel processes are in relation to substantially the same dispute, there is a broad principle of contractual interpretation:
There is every reason to presume that reasonable parties will wish to have the relationships created by their contract and the claims arising therefrom, irrespective of whether their contract is effective or not, decided by the same tribunal and not by two different tribunals. (Bundesgerichtshof's Decision of 27 February 1970, Juristische Zeitung 1970, 730; adopted in Fiona Trust v Privalov [2007] UKHL 40)
The presumption is not very strong overall, as for example in Enka v Chubb [2020] UKSC 38 at 199 there could be reasons why people would rationally depart from it. As noted by Mr Justice Andrew Smith in Lombard North Central v GATX [2012] EWHC 1067 (Comm) at 16, if
the parties have certainly agreed to refer to arbitration only certain disputes that might arise from their relationship [...] the risk of proceedings before both the courts and an arbitral tribunal is inherent in the agreement.
That decision has been approved by the Supreme Court in the recent case of Republic of Mozambique v Privinvest [2023] UKSC 32. The existence of that case shows how complicated things might get: this was a preliminary issue to the real dispute, which made it all the way to the UK's highest court. It is also an example of where the litigating parties were simultaneously engaged in arbitration (in fact multiple arbitrations, in Switzerland), and where part of their dispute concerned the scope of the arbitration agreements. Privinvest asked for a stay under section 9 but the court concluded that "a subordinate factual issue" in the context of "a multi-party litigation" (para 107) was not suitable to be sent to separate arbitration. The court thus allowed the main litigation to proceed to its next phases, even though there was related arbitration activity going on as well.
