There are two separate issues here; the pre-action protocol, and the limitation period.
I'm assuming that for the cause of action there is no specific pre-action protocol. If there is, then you should comply with that protocol. Otherwise, the general Practice Direction - Pre-action Conduct and Protocols (the "Protocol") applies.
The Protocol does not provide for specific procedures or hard deadlines, and says nothing at all about replying to a reply. Instead, it provides at paragraph 3 that:
Before commencing proceedings, the court will expect the parties to
have exchanged sufficient information to—
(a) understand each other’s position;
(b) make decisions about how to proceed;
(c) try to settle the issues without proceedings;
(d) consider a form of Alternative Dispute Resolution (ADR) to assist
(e) support the efficient management of those proceedings; and
(f) reduce the costs of resolving the dispute.
This is typically complied with by the claimant sending a letter which sets out, clearly and concisely, what is being claimed and the reasons why, and proposing how the claim can be resolved without going to court. The defendant then replies to provide the reasons why they dispute the claim. Provided that these two letters comply with the above rule then no further correspondence is necessary and the claimant can proceed with issuing the claim.
Paragraph 6 elaborates and provides some guidance as to timescales (emphasis mine):
Where there is a relevant pre-action protocol, the parties should
comply with that protocol before commencing proceedings. Where there
is no relevant pre-action protocol, the parties should exchange
correspondence and information to comply with the objectives in
paragraph 3, bearing in mind that compliance should be proportionate.
The steps will usually include—
(a) the claimant writing to the defendant with concise details of the
claim. The letter should include the basis on which the claim is made,
a summary of the facts, what the claimant wants from the defendant,
and if money, how the amount is calculated;
(b) the defendant responding within a reasonable time - 14 days in a
straight forward case and no more than 3 months in a very complex one.
The reply should include confirmation as to whether the claim is
accepted and, if it is not accepted, the reasons why, together with an
explanation as to which facts and parts of the claim are disputed and
whether the defendant is making a counterclaim as well as providing
details of any counterclaim; and
(c) the parties disclosing key documents relevant to the issues in
Note that a failure to comply with the Protocol will not affect the outcome of the case. Rather, it will have cost implications (i.e. make it more likely that the winner receives an adverse costs order).
If you have received a reply from the defendant and then take an excessive amount of time to issue a claim, you should probably send a fresh demand. The courts have a lot of discretion for costs orders and have been known to frown on proceedings being initiated after a long delay. In particular if the delay is long then the defendant may have forgotten about it or assumed that you had dropped the case. Evidence may be lost, memories fade, etc.
The steps you take under the Protocol have no affect whatsoever on the limitation period. Paragraph 17 re-iterates this and provides guidance on what to do if compliance with the Protocol would cause you to miss the deadline:
This Practice Direction and the pre-action protocols do not alter the
statutory time limits for starting court proceedings. If a claim is
issued after the relevant limitation period has expired, the defendant
will be entitled to use that as a defence to the claim. If proceedings
are started to comply with the statutory time limit before the parties
have followed the procedures in this Practice Direction or the
relevant pre-action protocol, the parties should apply to the court
for a stay of the proceedings while they so comply.
You mentioned a limitation period of 3 years. You should be aware that this is rather unusual and the usual limitation period for most types of claims is 6 years (e.g. see sections 2, 5, and 9 of the Limitation Act 1980 in relation to causes of action founded on tort, contract, and statute respectively). If you open the full Act and search for the term "three years" you will find the causes of action which have a 3 year limitation period (e.g. personal injury, negligence).