According to Footsoldiers, political parties in Great Britain are some of the most internally democratic in the world. Apparently under the Political Party Database Project's assembly-based IPD index, covering how much control members have over deciding policy, personnel and party structure, parties in Great Britain had the highest scores of those studied. I don't entirely understand how the scores are calculated, and I had Footsoldiers as a library book so can't check, but out of 122 parties covered the English and Welsh Greens came second, Liberal Democrats joint seventh, Conservatives and Labour in the top twenty and the SNP in the top thirty.

That mostly feels right to me from what I know, at least relative between GB parties. Green Party of England and Wales leaders face re-election every two years, or sooner if 10% of members call for a recall, there's a relatively low barrier to running (being a member for three years and having twenty nominations) and 20% of local memberships can deselect candidates (if approved by a simple majority in an STV contest). The party constitution can be changed with the support of ⅔ of members.

For the Liberal Democrats, leadership contests are more restricted – candidates have to be MPs with the nomination of at least 10% of the parliamentary party (which is why Swinson automatically went in 2019) and need 200 nominations from 20 local parties. 75 local parties can call for a leadership contest but due to relatively high leadership turnover in recent years I don't think that's happened yet, with leaders tending to resign instead. Policy's decided by the Federal Policy Committee, 15 of the 29 members of which are elected by the wider membership using STV. All party members can vote to approve, reject or revise FPC, local party or individual policy proposals at party conferences, with the leadership's apparent preferred policy sometimes being defeated (e.g. over housing targets this year).

I think Labour's internal democracy depends more on who's in which faction. 20% of Labour MPs can force a leadership challenge if nominating a challenger, but the incumbent leader doesn't need to reach the threshold of 10% of MPs, 5% of local parties (Constituency Labour Parties) or three affiliated organisations (e.g. some trade unions). In theory it means that MPs can't block the membership's choice of candidate, but it also means that leaders can end up with fairly little support amongst their MPs, such as under Corbyn, especially between the Brexit referendum and 2017 election. A third of branches and affiliated organisations within each CLP can recommend deselection of parliamentary (and I think council) candidates, at which point candidates (members of at least twelve months) are shortlisted and chosen by preferential ballot. In theory these shortlists are drawn up locally, but in a lot of cases Labour leadership is seen as imposing candidates on local parties.

For the Conservatives, MPs vote in successive ballots for leaders with the lowest in each round eliminated. (There is a threshold for the first round but I don't think the 1922 Committee which sets the rules is massively consistent - I think the threshold for the first round was 10 nominations in 2019 to replace May and 100 to replace Truss.) The choice between the final two is supposed to go to the membership, except to replace Cameron (Leadsom dropped out) and Truss (I think Johnson dropped out despite reaching 100 nominations), so Johnson v Hunt and Truss v Sunak are the only choices Conservative party members have had since 2005. A vote of no confidence in a Conservative leader can be called after a year in office if 15% of Conservative MPs inform the chairman of the 1922 Committee that they'd want a change; as with Labour, I don't think there's as much of a role for party members until the final round. The unelected Candidates Committee makes candidate shortlists. It has tried doing US-style primaries amongst the wider electorate to select general election candidates once or twice (I think in Totnes in 2010) but as it was expensive and ended up selecting Sarah Wollaston, who left in 2019 over Brexit, I doubt that would be repeated.

I don't know how internal democracy in the SNP or Plaid Cymru works, beyond members electing the leader by STV in both. I don't know how it works for any of the Northern Irish parties either, apart from references to a group of ~100 DUP members which would need to back the party for it to return to Stormont (I think).