Quick question, what do you mean by radical?
Radicals were something of a third force in British politics in the 19th century, dating from Charles Fox in the 1790s. Here's link to the
Wikipedia article. Chamberlain always considered himself a Radical at heart.
So your thinking that Churchill brings the Liberal Unionists into the Conservative Party? Or does he leave the Conservative Party to form a new party? If the first one happens, then you could see the Liberals torn apart by the "Home Rule" issue and Labor becoming the other major party.
The Liberal Unionists have already and in OTL aligned with the Conservatives to form a parliamentary majority in 1886, following the split in the Liberal Party over Home Rule for Ireland and the general election in June, 1886. Chamberlain was viewed with contempt for this action, since he had betrayed Radical principles in allying with the Tories, at least so though the Liberals. In truth, Chamberlain's principles were quite complicated and he continued to push for reform (land reform, education, local government, etc). The following is a summary of Chamberlain's association with the new Unionist government in 1886
Wikipedia said:
Unlike Hartington and the Whigs, Chamberlain did not enter the Unionist government, aware that the hostility to him in the Conservative ranks meant that an agreement with them could extend merely to Ireland. Not wishing to alienate his Radical support base, Chamberlain refrained from reaching a broader settlement. The Liberal mainstream cast Chamberlain as a villain, shouting "Judas!" and "Traitor!" as he entered the House of Commons chamber. Unable to attach himself decisively to either party, Chamberlain sought concerted action with a kindred spirit from the Conservative Party, Lord Randolph Churchill. In November 1886, Churchill announced his own 'Unauthorised Programme' at Dartford, the content of which had much in common with Chamberlain's own recent manifesto, including smallholdings for rural labourers and greater local government. Next month, Churchill resigned as Chancellor of the Exchequer over military spending, and when the Conservative mainstream rallied around Salisbury, Churchill's career was effectively put to an end, along with Chamberlain's hope of creating a powerful cross-party union of Radicals. The elevation of Goschen to the Treasury symbolised the increasingly close relationship between non-Radical Liberal Unionists and the Conservatives, thereby isolating Chamberlain further.
This makes it seem that R. Churchill's resignation was an attempt to gain the upper hand, which would have given Chamberlain more influence. As it was, the Unionists did pass some of the reforms he championed, particularly in education. Chamberlain was delighted: "I have in the last five years seen more progress made with the practical application of my political programme than in all my previous life. I owe this result entirely to my former opponents, and all the opposition has come from my former friends," he wrote in 1891. However, Chamberlain remained on the outskirts of Unionist government. He attained his most influence as Colonial Secretary during the outbreak of the Boer War. He remained distant from Salisbury and Balfour and in 1906 campaigned on a platform of Tarrif Reform (Imperial Preference, as it would become known). He was something of a maverick without influence over the Party itself. An association with R. Churchill, given Churchill's own Tory Democracy, might have produced a further shift.
This would be cool. So this party is the National Tory Party or what exactly is this party going to be called?
Indeed, it would be cool. If one is really devoted, it might even lead to an interesting Britwank. I think the Party's Name would be some what murky. It'd probably be left for some quiet quip to really give a lasting name. I'd guess it might formally be known as the (National) Unionists for at least some time. If Chamberlain has free reign and support, than Home Rule might be obviated by the introduction of National Councils for England, Wales, Scotland, and Ireland, making the Westminster Parliament a federal entity within the United Kingdom. An early introduction of Imperial Preference and Colonial/Imperial Conferences might lead to more institutional cohesion within the British Empire.
The party will, if taken to a full Chamberlainite perspective, have strong support amoung industrial workers, which may provide a new home for Trade-Unionsts, apart from the Liberal Party. Chamberlain favored reform of industry that may have induced Britain to make its business more efficient. All of these policies were controversial in Britain OTL, but they failed largely because Chamberlain advocated them outside of without the organ of a Party. Hence the controversy stemmed in part from the fact that they were maverick stands against Party government apart from being Radical plans to change the country.